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](https://mathoverflow.net/questions/21606/orbit-structures-of-conjugacy-class-set-and-irreducible-representation-set-under/21630#21630)). 
Can something more specific be said in this situation ? (e.g. Geoff Robinson mentions Glauberman correspondence [here](https://mathoverflow.net/questions/102879/bijection-between-irreducible-representations-and-conjugacy-classes-of-finite-gro/102896#102896)). 


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?](https://mathoverflow.net/questions/21200/brauers-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...



Sets M,N with G action such that C[M] = C[N] as G modules, how are they related ? 
An old paper of mine [On exponentiation of G-sets, Discrete Math. 135 (1994) 69-79] contains, among other things, the result that the free complex-linear actions generated by two $G$-sets $X$ and $Y$ are isomorphic if and only if the power sets, $P(X)$ and $P(Y)$, with the obvious induced actions of $G$, are isomorphic as $G$-sets. Curiously, the "intermediate" property, that the power sets are isomorphic as $(\mathbb Z/2)$-linear representations of $G$, is not equivalent to these.


