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

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.