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 $$\operatorname{Aut}(N)$$.

Similar facts

 * Automorphism group action lemma for quotients

Applications

 * Centerless and characteristic in automorphism group implies automorphism group is complete
 * Semidirectly extensible implies linearly pushforwardable for representation over prime field