Finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Definition
A finite group $$G$$ is said to have the same orbit sizes of conjugacy classes and irreducible representations under automorphism group if the following holds:

Let $$C(G)$$ be the set of conjugacy classes of $$G$$ and $$R(G)$$ be the set of irreducible representations (up to equivalence) of $$G$$ over $$\mathbb{C}$$. The automorphism group $$\operatorname{Aut}(G)$$ acts on both $$C(G)$$ and $$R(G)$$. The condition we need is that the orbit sizes in $$C(G)$$ and $$R(G)$$ be equal.