Automorphism group action lemma
Suppose is a group, and are subgroups such that . Suppose is an automorphism of such that the restriction of to gives an automorphism of , and such that also restricts to an automorphism of , say . Consider the map:
that sends an element to the automorphism of induced by conjugation by (note that this is an automorphism since ). Then, we have:
where denotes conjugation by in the group .