Application of Brauer's permutation lemma to group automorphism on conjugacy classes and irreducible representations

Statement
Suppose $$G$$ is a finite group. Suppose $$\sigma$$ is an automorphism of $$G$$. Then, $$\sigma$$ induces permutations both on the set of conjugacy classes $$C(G)$$ of $$G$$ and on the set of equivalence classes of irreducible representations $$I(G)$$ of $$G$$ over a splitting field. Both these automorphisms have the same cycle type.

Related facts

 * Application of Brauer's permutation lemma to Galois automorphism on conjugacy classes and irreducible representations