You should not pass the ID Token that you receive to Google back to the app to use as an auth token. The ID Token will have an audience specific to your app, not the ultimate client.
If you just passed this back to clientA, then you would open it up to auth attacks where clientB could obtain an Google ID Token via your auth service and then use that token to gain access to user data in clientA.
A much better approach would be to mint your own ID Token and provide an audience designation for each app that uses your service.
Monitoring session state is not as straightforward. The easiest thing to do is for your service to wrap G+ sign-in libraries and interface with the session state api's.