Application of Brauer's permutation lemma to group automorphisms on conjugacy class-representation duality

Statement
Suppose $$G$$ is a finite group and $$K$$ is a splitting field for $$G$$. Any automorphism $$\sigma$$ of $$G$$ induces a permutation $$\sigma_1$$ on the set $$C(G)$$ of conjugacy classes of $$G$$, and a permutation $$\sigma_2$$ on the set of irreducible representations (up to equivalence) of $$G$$.

Our statement is that the two permutations $$\sigma_1$$ and $$\sigma_2$$ have the same cycle type.

Similar facts

 * Application of Brauer's permutation lemma to Galois automorphisms on conjugacy class-representation duality
 * Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Opposite facts

 * Number of orbits of irreducible representations need not equal number of orbits of conjugacy classes under automorphism group

More on Brauer's permutation lemma

 * Brauer's permutation lemma
 * Analogue of Brauer's permutation lemma fails over rationals for every non-cyclic finite group
 * Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals

Facts used

 * 1) uses::Brauer's permutation lemma