Follow these instructions if you are currently using a Google Groups email alias as your shared inbox. Note: you must be a Google Apps administrator to make these changes.
1. Log in to your Google Apps Admin Console (https://admin.google.com/) and expose more controls by clicking the Google Groups icon at the bottom.
2. Find the group that you want to add to Inbox and then choose Manage users in Info.
3. The best way to set up Inbox is to remove all of the existing members, and enter the Inbox forwarding email address (i.e. firstname.lastname@example.org).
Google Groups also per default adds a footer to every outgoing email. To disable, go to Group Settings in the upper right - then back on the left under Settings, click Email options and deselect the Email footer option.
After you complete this step you're finished with the Google part.
4. Go back to your Inbox and click "Test forwarding".
This will send a special email to your group alias to verify that everything is set up correctly. This process can take up to a minute to complete.