Just use the Rules UI to import the rule (in Rules Export format) in your own site:

    { "rules_notification_about_new_group_invitation" : {
        "LABEL" : "Notification about new group invitation",
        "PLUGIN" : "reaction rule",
        "OWNER" : "rules",
        "REQUIRES" : [ "rules", "group" ],
        "ON" : { "group_membership_insert" : [] },
        "IF" : [
          { "data_is" : { "data" : [ "group-membership:status" ], "value" : "invited" } }
        ],
        "DO" : [
          { "drupal_message" : { "message" : "The user with name [group-membership:user] just got invited by [group-membership:invited-by:name] to join the group named [group-membership:group:title] and with group type [group-membership:group:group-type:label]." } }
        ]
      }
    }

This rule will show a drupal message, each time a user is invited to join some group, similar to this sample message:

> The user with name Demo.User just got invited by Pierre.Vriens to join the group named Using the Answers module and with group type Class.

All that's left to do is to replace the Rules Action (to show a Drupal message) by an appropriate eMail (= a kind of Rules 101, right?).

Note that for some user who is just "*added*" to a group (instead of "*invited*") by an authorized user to do so, such message will not be shown. That's because of the Rules Condition included: the status for the selected group is not "invited" in that case.