Centralizer of coprime automorphism in homomorphic image equals image of centralizer

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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 \varphi is an automorphism of G whose order is coprime to the order of G. Suppose N is a normal \varphi-invariant subgroup of G, and let \pi:G \to G/N denote the quotient map. Then, \varphi has a natural induced action on the quotient group G/N, and we have:

\pi(C_G(\varphi)) = C_{G/N}(\varphi).

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