It seems still unclear. If g is a permutation of a set X of whatever nature,
S a subset of X, then S^g is a set, otherwise one would expect a message
like "Error, <set> must be a set in ...OnSets".
That is, S^g is uniquely defined. That is , S^g should be a set, so
the appropriate flag of it is 'yes'.
Certainly I agree that there are operations which can disrupt the order of
S, say as it was pointed out by making changes to elements,
but S^g should be exempt from this list.
The problem is that the 'set' object cannot know whether one of its
memebrs has been altered. Accordingly, the flag can NEVER be used on
a list containing changeable objects.