@KimMonaghan ...
To expand on what @chriscas said... you don't put students into "Group Sets". You can put students into "Groups" within those "Group Sets". I do agree with Chris that you can move students from one "Group" to another "Group" within the same "Group Set"...keeping in mind the potential consequences of doing so.
There really isn't option of moving a student from one "Group Set" to another...because you can add a student to a "Group" within a "Group Set" .... and then in a completely different "Group Set", you might have that same student added to a different "Group". This student can be added to both "Groups" in both "Group Sets" at the same time. For example, you might have a "Group Set" called "Group Project 1", and another "Group Set" called "Group Project 2". You could assign a student to a "Team Red" Group in the first project and then to "Team Blue" in the second project.
I hope this helps to clarify things a bit...and hopefully I haven't muddied the waters too much.