Regardless of the setting issue, answering this specifically, in general:
Is it possible to Delete the Group (freeing the name up) and re-create the Group?
Yes, you can always delete and recreate a group, but before doing so, always change the group name (address) first to something unique, like let's say Dan + 3 first digits of your SSN + a relative's name + last 4 digits of your phone, you get the idea, then delete the group; if you don't do this, the name won't be available for reuse for possibly quite some time and you'll be stuck until it becomes available.
