Semidirect Product with Trivial Action is Direct Product

Theorem
Let $H$ and $N$ be groups.

Let $\operatorname{Aut}(N)$ denote the automorphism group of $N$.

Let $\phi : H \to \operatorname{Aut}(N)$ be defined as $\phi(h) = \operatorname{id}_N$ for all $h\in H$.

Let $N\rtimes_\phi H$ be the corresponding semidirect product.

Then $N\rtimes_\phi H$ is the direct product of $N$ and $H$.