Using google authentication and google groups in my code

Question

I want to use Google authentication to authenticate a user, and then, hopefully, using the Google APIs to find out if this user is a member of a specific group. If so, my program starts, otherwise the program will allow the user to take some action. As a google group manager, I can manually add / remove users from google.

I read and understand the OAuth part. Therefore, I know that I can allow the user without problems. The problem is confirming group membership.