Having to edit a group to change its grouping is a bit less effective/usable. Considering we have a page that does everything else grouping <-> group-wise. Users should be able to move groups around from one grouping to another (and to none) without having to edit the group. This can be done using the following method:
1. Click on the group
2. Click on a "Move to Grouping" button
3. Click on the destination grouping
4. Click on a "Select Grouping as Destination" button
With more AJAX, this can be reduced to 3 steps, or even 1 using drag and drop.
An additional "Make group Orphan" button/option is also needed.