Can an aerodrome in class G airspace be controlled?
An airport in class G airspace can have an operating control tower. This is a case of a towered airport in uncontrolled airspace. The airspace is uncontrolled but tower communication must be established within a certain distance and for use of the airports runways.

There are applicable regulations that address this situation. 14 CFR §91.126 (d) requires pilots to establish and maintain communication with a Class G towered airport before entering a 4 NM radius below 2500 AGL.

As of today, Lake City, Florida (KLCQ) is an example of such an airport, although that will change at some point in the future.