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:

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

Related facts

Applications

• Stability group of subnormal series of finite group has no other prime factors
• Centralizer-commutator product decomposition for finite groups
• Burnside's theorem on coprime automorphisms and Frattini subgroup

References

Textbook references

• Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, Page 17, Theorem 1.6.2, ^{More info}