Centralizer-commutator product decomposition for finite groups

This article states and (possibly) proves a fact that involves two finite groups of relatively prime order, requiring the additional datum that at least one of them is solvable. Due to the Feit-Thompson theorem, we know that for two finite groups of relatively prime orders, one of them is solvable. Hence, the additional datum of solvability can be dropped. However, the proof of the Feit-Thompson theorem is considered heavy machinery.
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This article states and (possibly) proves a fact about a finite group and a Coprime automorphism group (?): a subgroup of the automorphism group whose order is relatively prime to the order of the group itself.
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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:

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

Facts used

1. 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. Centralizer of coprime automorphism group in homomorphic image equals image of centralizer if either is solvable