Due to the reimplementation of user group pages for 6.1 an option that existed in previous versions has disappeared. The possibility to give the permission to manage the pages of the user group to a user other than the owner of the user group. When trying to assign that permission it simply does not exist as part of the actions of a user group.
The proper fix of the issue should allow for proper delegation of the permissions, including all the staging permissions or permissions on individual pages. To that end, the best solution would be to expose the pages of a user group as a user group site which has its own set of permissions similar to those of any site.
Once this solution has been implemented, the steps to verify that it is working fine are:
1. Create a user group and add some pages to it
2. Create a role and give it the permissions to view the user groups tool in the control panel
3. Go to the user groups tool and click Actions > Site Permissions. Give the "Manage pages" and "View" permissions to the role that was just created
3. Create a user with that role and who is a member of the user group and log in as that user
4. Access the user groups tool and use the actions menu of the user group. An option to manage the pages should be shown
5. Click it and try adding pages and updating existing ones