In this post, the answer mentions:
Now I leave you to check the following isomorphisms: \begin{align} \mathbb Z_{11}\rtimes_{\varphi_1}\mathbb Z_5&\to\mathbb Z_{11}\rtimes_{\varphi_2}\mathbb Z_5\\ (n,h)&\mapsto (3n,h) \end{align}
\begin{align} \mathbb Z_{11}\rtimes_{\varphi_1}\mathbb Z_5&\to\mathbb Z_{11}\rtimes_{\varphi_3}\mathbb Z_5\\ (n,h)&\mapsto (9n,h) \end{align}
\begin{align} \mathbb Z_{11}\rtimes_{\varphi_1}\mathbb Z_5&\to\mathbb > Z_{11}\rtimes_{\varphi_4}\mathbb Z_5\\ (n,h)&\mapsto (5n,h) \end{align}
When I attempted to solve this problem, I reached this very last bit, where I got all of these semi-direct products, but didn't know how to efficiently (or at all) check whether they are isomorphic.
How could I have gone about identifying that these semi-direct products are isomorphic? (also - is there some general approach for this? I tend to get stuck at this stage when practising similar questions). I should note -- I am looking for a method for this, and not a higher level theorem that can be applied.
I did consider trying to assume I have found a mapping between the generators of form $(n,1), (0,1)$, i.e., assuming $(n,1) \mapsto (kn,1)$ for some $k$ and $(0,1) \mapsto (0,1)$ but it got quite messy before I reached a solution (a $k$ that satisfies this being a bijective homomorphism).
