How to send an eMail to registered users after they are invited using the Group module?
You can use the Rules module to achieve that "people who already have a user account, also receive an email invitation in addition to the invitation on their user account" (as in your question).

Here is the rule (in Rules export format) to actually do so:

{ "rules_send_email_after_inviting_a_registered_user" : {
    "LABEL" : "Send eMail after inviting a registered user",
    "PLUGIN" : "reaction rule",
    "OWNER" : "rules",
    "REQUIRES" : [ "rules", "group" ],
    "ON" : { "group_membership_insert" : [] },
    "IF" : [
      { "data_is" : { "data" : [ "group-membership:status" ], "value" : "invited" } },
      { "NOT user_has_role" : {
          "account" : [ "group-membership:user" ],
          "roles" : { "value" : { "1" : "1" } }
        }
      }
    ],
    "DO" : [
      { "mail" : {
          "to" : "[group-membership:user:mail]",
          "subject" : "You have a pending group invitation",
          "message" : "Hi there!\r\n\r\n[site:current-user:name] has invited you to become a member of the group \u0022[group-membership:group]\u0022 on [site:name].\r\nIf you wish to accept the invitation, you need to login with your existing account first, which is \u0022[group-membership:user:name]\u0022.\r\nPlease visit the following address in order to do so: [site:login-url]\r\n\r\nKind regards,\r\nThe [site:name] team",
          "from" : "[site:current-user:mail]",
          "language" : [ "" ]
        }
      },
      { "drupal_message" : { "message" : "You just invited existing user with name [group-membership:user:name] and eMail Id [group-membership:user:mail] to the group named [group-membership:group].\r\n\r\nAn appropriate message about this invitation just got eMailed to [group-membership:user:name] also." } }
    ]
  }
}


Some more details about the magic included in this rule:


Sending an invitation in the Group module's terminology is equivalent to inserting a new Group Membership with status "invited" (how easy can it be?). So that explains the Rules Event I'm using, and the first Rules Condition.
Since we only want to send such eMail to users who already have an account, the 2nd Rules Condition is to check that the invited user does NOT (!!!) have role anonymous.
The Rules Action is just to send an appropriate eMail, using specific tokens related to the Group Membership, etc.
For debugging purposes, the 2nd Rules Action shows a confirmation message about the triggered eMail (feel free to remove that action if you don't like it).


Just import the above rule (using the Rules UI) in your own site, with no customization (changes) needed to see it at work in your own site (don't worry, there is no spyware in the eMail being produced ...).

The eMail content is pretty close to the eMail delivered when inviting anonymous users (the only difference is that it has an instruction to login with their existing account, also included in the eMail body). Here is how a sample eMail body looks like:

Hi there!

Pierre.Vriens has invited you to become a member of the group
"Drupal for Sitebuilders" on www.example.com.

If you wish to accept the invitation, you need to login with your existing
account first, which is "Dries.Buytaert".

Please visit the following address in order to do so:
http://www.example.com/user

Kind regards,
The www.example.com team


Possibly you may want to finetune the eMail subject, or eMail body. But apart from that no other changes should be needed.

Voilà ...