let *G* be a [EDIT: FINITE] group. Suppose *C* is the set of conjugacy classes of *G* and *R* is the set of (equivalence classes of) irreducible representations of *G* over the complex numbers.

The automorphism group of *G* has a natural action on *C* and also on *R* (we can make both of these left actions). My questions:

- Under what conditions are
*C*and*R*equivalent as $\operatorname{Aut}(G)$-sets? This is definitely true, for instance, if every automorphism is inner, if the outer automorphism group of*G*is cyclic (it then follows from Brauer's permutation lemma) and it is also true if the quotient of the automorphism group by the group of class-preserving automorphisms is cyclic (again by Brauer's permutation lemma). But it also seems to be true in a number of other cases, such as the quaternion group, where the outer automorphism group is a symmetric group of degree three. - A weaker condition: under what conditions are the orbit sizes under $\operatorname{Aut}(G)$ for
*C*and*R*the same?