Centralizer-commutator product decomposition for finite groups and cyclic automorphism group

Statement
Suppose $$G$$ is a finite group and $$H$$ is a cyclic subgroup of $$\operatorname{Aut}(G)$$. Let $$[G,H]$$ be the subgroup of $$G$$ generated by all elements of the form $$g\sigma(g)^{-1}$$ for $$\sigma \in H$$. Let $$C_G(H)$$ denote the subgroup of $$G$$ comprising those elements fixed by every element of $$H$$. Then:

$$G = [G,H]C_G(H)$$.

Related facts

 * Centralizer-commutator product decomposition for Abelian groups
 * Centralizer-commutator product decomposition for finite nilpotent groups

Facts used

 * 1) uses::Centralizer of coprime automorphism in homomorphic image equals image of centralizer: Suppose $$\varphi$$ is an automorphism of $$G$$ of order coprime to the order of $$G$$. Suppose $$N$$ is a normal $$\varphi$$-invariant subgroup of $$G$$. Then if $$\pi:G \to G/N$$ denotes the quotient map, we have $$\pi(C_G(\varphi)) = C_{G/N}(\varphi)$$.