WordPress Integration with Google Groups

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Most of your question is off topic, I guess, because it involves a question about Google’s APIs. You would need three things:

  1. Create the possibility to login to your site with a Google account. There are plugins that make this possible, like this one.
  2. Import the list of group members from Google. There probably is an API for this, but that is out of the scope of this site. Once you have an array of users an mailaddresses, you can use wp_create_user to make new entries in your user table. You would have to set a transient for this, to prevent looking for new group members at Google with every pageload. Also, you should create a special user role for group members.
  3. Finally, in your templates you can add a check for users that are logged in with that special role to display exclusive information.