Automorphism group has linearly equivalent actions on set of conjugacy classes and set of irreducible representations
Suppose is a finite group, is the set of conjugacy classes of and is the set of equivalence classes of irreducible representations of over . The automorphism group acts as permutations on the sets and . The claim is that these permutation actions, when viewed as linear representations over , are linearly equivalent.
Note that since linearly equivalent not implies permutation-equivalent (the failure of the analogue of Brauer's permutation lemma for non-cyclic groups) the representations need not be equivalent as permutation representations.