Commutator of finite group with coprime automorphism group equals second commutator

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

$$[[G,H],H] = [G,H]$$.