Commutator of a group and a subgroup of its automorphism group is normal

Statement
Suppose $$G$$ is a group and $$A$$ is a subgroup of the automomorphism group of $$G$$. Then the commutator:

$$[A,G]$$

defined as the commutator of the subgroups $$A$$ and $$G$$ inside the semidirect product $$G \rtimes A$$, is a normal subgroup inside $$G$$.

Facts used

 * 1) uses::Subgroup normalizes its commutator with any subset