The most common use of groups is probably to define access rights for non-public categories. Currently the two things - creating a group and creating a category - are entirely separate steps and since the name of a category is often the same as the name of the group that has access rights, it in a way feels like doing the same thing twice.
How about adding a check box to the ”create new group” page:
[ ] Create a private category for this group
When ticked, creating the group will - you guessed it - automatically create a private category with the same name as the group and limit create, reply, see rights to members of the new group.
A more sophisticated version of this feature would allow the user to change the default category name already on the group page, but the basic implementation with just the default name would also be a great improvement.