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

Statement
Suppose $$G$$ is a finite group and $$r$$ is an integer relatively prime to the order of $$G$$. Suppose $$K$$ is a field and $$L$$ is a splitting field of $$G$$ of the form $$K(\zeta)$$ where $$\zeta$$ is a primitive $$d^{th}$$ root of unity, with $$d$$ also relatively prime to $$r$$ (in fact, we can arrange $$d$$ to divide the order of $$G$$ because sufficiently large implies splitting). Suppose there is a Galois automorphism of $$L/K$$ that sends $$\zeta$$ to $$\zeta^r$$. Consider the following two permutations:


 * The permutation on the set of conjugacy classes of $$G$$, denoted $$C(G)$$, induced by the mapping $$g \mapsto g^r$$.
 * The permutation on the set of irreducible representations of $$G$$ over $$L$$, denoted $$I(G)$$, induced by the Galois automorphism of $$L$$ that sends $$\zeta$$ to $$\zeta^r$$.

Then, these two permutations have the same cycle type. In particular, they have the same number of cycles, and the same number of fixed points, as each other.