To reset a user account with a Gmail suspension:

  1. Sign in to the Google Apps administrator control panel.
     
  2. Click the Organization & users tab.

  3. Find the user account suspended from Gmail. You can either search or browse:
    • Search: Enter the user name in the search box at the top of the Google Apps administrator control panel, then click Search accounts.
    • Browse: Click the Organization & users tab from the top menu. In some cases, the Status column displays a yellow caution symbol identifying the user has been suspended from Gmail. 

  4. Click the suspended user to see User information.

  5. In the yellow bar near the top of the page, note the reason for suspension and click the Reset user button to re-enable Gmail for the user.

    Reset users can sign in and regain full access to the Gmail inbox. Please note, each reset appears in the control panel audit log.
     
  6. Click the accompanying Learn more link to find out how the user can avoid this limit in the future. 
If the user account has a Gmail suspension and the reset button is not available, the user must wait 1-24 hours before the limits are automatically reset. It is still possible the user to sign into and use all other Google services (e.g. Google Calendar, Google Docs).

For additional detail, check out the Google provided resource .