I am doing my thesis about semidirect products mainly and wondering how to solve the following question:
Let $G$ be a semidirect product of a normal subgroup $N$ with two elements by a subgroup $H$. Show that $G$ is an internal direct product of $N$ and $H$.
(I know that a semidirect prouct of $N$ by $H$ is the direct product if and only if the homomophism $H$ to $\operatorname{Aut}(N)$ is trivial, that is $\operatorname{id}(N)$ for all $h \in H$, but I dont really know how to move forward.)