Automorphism group has linearly equivalent actions on set of conjugacy classes and set of irreducible representations

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Statement

Suppose G is a finite group, C(G) is the set of conjugacy classes of G and R(G) is the set of equivalence classes of irreducible representations of G over \mathbb{C}. The automorphism group \operatorname{Aut}(G) acts as permutations on the sets C(G) and R(G). The claim is that these permutation actions, when viewed as linear representations over \mathbb{Q}, 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.

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