As a group member, no moving data to a group that you're not a member of
As a group owner, no moving other's data to a group that you're not an owner of
As anyone, (partially) restrict move to group with lower permissions.
Future (advanced features)
As a group member, move another members data to a group that both are a member of.
Should moving to the "user" group be allowed? The primary issue is one of testing. Are there any dead-ends that the data gets into, so that it can't get back out? (i.e. once someone links to an image in "user", then it could get stuck)
Should users be able to move data from group A to B while logged into C? At the moment, they can't, but the ChgrpI implementation could login into A automatically (and temporarily)