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.