Centralizer-commutator product decomposition for finite groups

Statement
Suppose $$G$$ is a finite group and $$H$$ is a subgroup of $$\operatorname{Aut}(G)$$ such that the orders of $$G$$ and $$H$$ are relatively prime. Then, we have:

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

Related facts

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

Facts used

 * 1) uses::Coprime implies one is solvable: This notorious corollary of the Feit-Thompson theorem states that given two finite groups whose orders are relatively prime, at least one of them is solvable.
 * 2) uses::Centralizer of coprime automorphism group in homomorphic image equals image of centralizer if either is solvable