Description of my issue:
If the update to Joomla is applied after the Akeeba Backup 3.11.1 update is applied, a 403 error page is returned when trying to apply the Joomla update. If the Joomla update (in this case from 2.5.20 to 2.5.21) is applied prior to applying the Akeeba Backup 3.11.1 update, the Joomla update succeeds. The prior version of Akeeba Backup is 3.11.0.
There are already a couple of tickets with this same problem open for the Joomla 3.3.1 update, I just wanted to confirm for your troubleshooting, the same problem exists with Joomla 2.5.x.