# Finite group whose automorphism group has equivalent actions on the sets of conjugacy classes and irreducible representations

## Definition

Suppose $G$ is a finite set, $C(G)$ 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 on the sets $C(G)$ as well as $R(G)$. We say that $G$ is a finite group whose automorphism group has equivalent actions on the sets of conjugacy classes and irreducible representations if the permutation representations of $\operatorname{Aut}(G)$ on $C(G)$ and $R(G)$ are equivalent, i.e., $C(G)$ and $R(G)$ are equivalent as $\operatorname{Aut}(G)$-sets.