# Centralizer of coprime automorphism in homomorphic image equals image of centralizer

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:

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