# Automorphism group action lemma

## Statement

Suppose $H$ is a group, and $N,G \le H$ are subgroups such that $G \le N_H(N)$. Suppose $\sigma'$ is an automorphism of $H$ such that the restriction of $\sigma'$ to $N$ gives an automorphism $\alpha$ of $N$, and such that $\sigma'$ also restricts to an automorphism of $G$, say $\sigma$. Consider the map:

$\rho: G \to \operatorname{Aut}(N)$

that sends an element $g \in G$ to the automorphism of $N$ induced by conjugation by $g$ (note that this is an automorphism since $G \le N_H(N)$). Then, we have:

$\rho \circ \sigma = c_\alpha \circ \rho$

where $c_\alpha$ denotes conjugation by $\alpha$ in the group $\operatorname{Aut}(N)$.