Consider two sets M, N such that C[M] is isomorphic to C[N] as representations of G, somewhat surprisingly it does not imply M, N are isomorphic as G sets. (Everything is finite.)

However it does imply that 1) |M| = |N| 2) number of orbits in M = number of orbits in N, because orbits correspond to invariant functions.

**Question**: Are there some other implications of C[M]=C[N] for the actions structure on M, N and corresponding module structure C[M]=C[N]? E.g. some numerical inequalities on sizes of orbits (there sizes can be different but may be not too much ?), stabilizers of points (can be non-conjugated, but may be somehow related), .... ?

As Benjamin Steinberg suggests number of orbits in MxMxM...xM (any times) will be the same as number of orbits in NxNxN...N, since these are dims of invariant functions in $C[M]^{\otimes L}$. Can it be the characterization ?

As far as I understand sets of conjugacy classes and irreducible representations of G provide such sets M,N for Aut(G). (hope I am correct ? See first sentence here). Can something more specific be said in this situation ? (e.g. Geoff Robinson mentions Glauberman correspondence here).

PS

**Question** What are natural examples/constructions/classifications of such sets ?

PSPS

The very related MO question by Vipul Naik is Brauer's permutation lemma -- extending to some other finite groups? and Ben Webster's answer with reference to his paper on the subj... But somehow I do not see the answer there...