Suppose we have $\varphi,\psi : H \to \operatorname{Aut}(N)$, so that $\varphi = \operatorname{Ad}_g \circ \psi$ where $\operatorname{Ad}_g:N\to N$ is given by $n\mapsto gng^{-1}$ for some $g \in N$.
Prove that $N \rtimes_{\varphi} H \cong N \rtimes_{\psi} H $.
I understand it's not the same conditions as here: When are two semidirect products isomorphic?
