Centralizer of coprime automorphism in homomorphic image equals image of centralizer

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:

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

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