Consider the isomorphism $$\begin{align*} \psi : G \rtimes_{\phi_{1}} H \to G \rtimes _{\phi_{2}} H \end{align*}$$ where $G \rtimes_{\phi_{i}} H$ is the semi-direct product of $G$ and $H$ with homomorphism $\phi_{i}: H \to \text{Aut}(G)$ .
Then can I have $\psi =(f, \pi)$ where $f \in \text{Aut}(G)$ and $\pi \in \text{Aut}(H)$ such that $$\begin{align*} \psi(g,h) = (f(g),\pi(h)) \end{align*}$$ That is, can I break down $\psi$ componentwisely?