In the US, in actual practice, workload permitting, will ARTC facilities grant SVFR clearance for surface-level Class E "extensions" (E4 airspace)?
In the US, in actual practice, will ARTC Centers and Approach Control
facilities grant clearance for Special VFR operations in E4
surface-level Class E &quot;extensions&quot;?

Some will, and some won't.

Does the answer vary between different ARTC facilities?

Yes.
Note the range of comments below obtained from questions posed by telephone in May 2019 to various ARTC Center and Approach Control staffers-- in most cases, the &quot;Watch Officer&quot; on duty, or else an &quot;Airspace Specialist&quot;--
(The summaries below are based on written notes and are not verbatim transcriptions.  The actual airspace descriptions are not from the staffers but rather are from the FAA's &quot;Airspace Descriptions And Reporting Points&quot; document, JO 7400.11E.)
Note also that in no case did the staffer indicate that the specific reason that SVFR would not be approved for an surface-level Class E &quot;extension&quot; was because it was designated as E4 airspace rather than E2 airspace.  Airspace definitions for some airports are actually written so that the surface-level Class E (E4) &quot;extensions&quot; convert to E2 surface-level Class E airspace when the control tower is closed, but no staffer ever indicated that that change in status would have any effect on whether a Special VFR clearance might be granted for that airspace.  Several staffers were not even aware that surface-level Class E airspace was broken down into several different subcategories (e.g. E2 versus E3/E4) in the FAA's &quot;Airspace Designations and Reporting Points&quot; document.
Facility-- Harrisburg Approach (PA)
Airport: KLNS-- Lancaster-- PA
Airspace description-- Part-time Class D, turns to surface-level Class E (E2) when tower closes.  Surface-level Class E (E4) &quot;extension&quot; to SE, converts to surface-level Class E (E2) when tower closes.
Staffer stated that when tower is closed, Harrisburg Approach may issue SVFR clearance for the entire &quot;surface area&quot;, which does include the airspace in the &quot;extension&quot;, because it  goes down to the surface, whether the tower is open or closed.  In response to my questioning, said it has nothing to do with whether the airspace is E2 or E4-- it's all &quot;Echo&quot; to them.  Said tower is only authorized to issue SVFR clearance for the Class D airspace, but if tower contacted Harrisburg Approach with a request, odds are good a SVFR clearance could be granted for the E4 extension as well.  Stated tower has to call Harrisburg Approach anyway when granting a SVFR clearance so it would not be a burden to ask.  Noted that S37 is within the E4 extension so a pilot flying from KLNS to S37 might have good reason to want a SVFR clearance including the extension.

Facility-- Seattle Center ARTCC
Airport-- KACV-- Arcata CA
Airspace Description-- surface-level Class E (E2) circle around airport effective 24/7, surface-level Class E (E4) &quot;extension&quot; to SE effective 24/7, depicted as separate areas on the sectional chart--
Staffer said a SVFR clearance could indeed include the entire Class-E-to-surface area.  In response to my questioning, stated made no difference whether the airspace was E2 or E4.

Facility-- Salt Lake City ARTCC
Airport-- KSHR --Sheridan County-- WY
Airspace description-- surface-level Class E (E2) airspace, effective 24/7, extends far to NW and SE of airport
Staffer's initial response was that a SVFR clearance would be limited to a certain radius from the airport such as 5 miles, but after taking a closer look at the sectional chart said a SVFR clearance could indeed include the entire Class-E-to-surface area.  Stated that since the area is quite large, let ARTCC know if you only need the northern part or the southern part.
The staffer also offered opinions on the airports below, while noting that they are not actually in the Salt Lake City ARTCC area of responsibility--
Airport-- KBIH -- Bishop-- CA
Airspace Description-- surface-level Class E (E2) circle around airport effective 24/7, surface-level Class E (E4) &quot;extensions&quot; to N effective 24/7, all depicted by a single dashed magenta line on the sectional chart--
Staffer said a SVFR clearance could indeed include the entire Class-E-to-surface area. In response to my questioning, stated made no difference that extensions were E4 not E2.
Airport-- KTVL-- South Lake Tahoe-- CA
Airspace Description-- surface-level Class E (E2) circle around airport effective 24/7, surface-level Class E (E4) &quot;extension&quot; to N effective 24/7, depicted as separate areas on the sectional chart--
Staffer said a SVFR clearance could indeed include the entire Class-E-to-surface area.  In response to my questioning, stated made no difference that extensions were E4 not E2.

Facility-- Minneapolis ARTCC
Airport-- KSTC-- St. Cloud MN
Airspace description-- Part-time Class D, turns to surface-level Class E (E2) when tower closes.  Surface-level Class E (E4) &quot;extension&quot; to SE, converts to surface-level Class E (E2) when tower closes.
Staffer stated that tower could only issue a SVFR clearance for the Class D airspace, but if tower were closed and pilot contacted Center, a SVFR clearance might be able to include the extension to SE.  Wasn't sure about this, said no one had ever asked.  Noted that there is small airport just inside the far end of the extension so there might be reason to desire such a clearance.  Was unfamiliar with designations E2 and E4 and planned to do more research on the topic.

Facility-- Oakland Center
Airport-- KBIH-- Bishop CA
Airspace Description-- surface-level Class E (E2) circle 5 miles in diameter around airport effective 24/7, surface-level Class E (E4) &quot;extensions&quot; to north effective 24/7, all depicted by a single dashed magenta line on the sectional chart--
(Staffer indicated that the above description was only valid for the winter season.  Stated that the summer configuration was class G up to 1200 feet.  That has apparently now changed, as the 2-25-21 Chart Supplement simply says &quot;AIRSPACE: CLASS E&quot;)
Regarding the winter configuration, staffer said a SVFR clearance would not be granted beyond 5 miles of the airport because the purpose of a SVFR clearance is just to get pilots up away from the airport or down into the airport, and the terrain to the north is mountainous so they don't want pilots flying SVFR there.  Also stated that the purpose of the extensions is to facilitate instrument arrivals and departures so they wouldn't want pilots flying SVFR there.  Said even in a case where terrain were flat they likely would not grant SVFR beyond a certain radius from the airport conforming to the basic inner Class D or Class-E-to-surface circle, because the extensions are to facilitate instrument arrivals and departures.  In response to my questioning, said it had nothing to with whether the airspace in the extensions was E2 or E4.

Facility-- Kansas City ARTCC
Airport-- KTOP-- Phillip Billard near Topeka KS
Airspace description-- Part-time Class D, turns to surface-level Class E (E2) when tower closes.  Surface-level (E4) &quot;extension&quot; to SE effective 24/7.
Staffer stated that a SVFR clearance could not include the extension because the purpose of a SVFR clearance is just get to pilots up away from the airport or down into the airport and if you can't do it within the D circle you should file IFR or not fly.  Stated that the purpose of the extension is to protect IFR traffic, so they wouldn't want to issue a SVFR clearance there.  In response to my questioning, stated made no difference that the airspace in the extension was E4 rather than E2 -- they just call it &quot;Echo&quot; and don't make that distinction.  Comments were not specific only to KTOP but also stating the general principle involved.

In essence, whether or not SVFR clearance will ever be granted by ARTC for a specific chunk of surface-level Class-E airspace that projects some distance from the basic circle of surface-level controlled airspace around an airport, seems very much to depend on the &quot;culture&quot; of the particular ARTC Center involved, as well as factors such as terrain hazards specific to the particular geographic location involved.

Note that the related answer Does an SVFR clearance extend to Echo surface extensions? quotes a 2010 letter from the FAA ATO Western Service Center stating that their opinion is that the FARs do not permit SVFR to be authorized in E4 surface-level Class E &quot;extensions&quot;.  The original query that provoked the answer was apparently specific to KSTC, an airport where all the surface-level controlled airspace reverts to Class G airspace when the tower is closed, so all requests for SVFR would always be directed to the tower, and ARTC would not be used to receiving any requests for SVFR for any of the airspace associated with that airport during the hours that the tower was closed.  Yet the letter is phrased more broadly and indicates as a general principle that SVFR should not be permitted in any E4 surface-level Class E &quot;extensions&quot;.
Clearly, not all ARTC facilities follow this guidance.

See these other related ASE questions and answers--
Does FAR 91.155c apply to class E surface extensions?
What indication has the FAA given that phrases like &quot;surface area of Class E airspace designated for an airport&quot; do or don&#39;t include E4 &quot;extensions&quot;?
What are the historical precedents of today&#39;s E2 and E4 airspace? (US)
Which parts of class E airspace can an ultralight (part 103) fly in without prior ATC authorization?