For example, I want to understand what are different $S_5 \rtimes \langle c\rangle_2$ products.
$\mathrm{Aut}(S_5)=\mathrm{Inn}(S_5)\simeq S_5$, so we can have direct product or $\psi: с \rightarrow \tau \in S_5$, such that $o(\tau ) = 2$.
I know that:
Let $N,H$ be groups, $ϕ:H\to\mathrm{Aut}(N)$ be a homomorphism, $\psi\in \mathrm{Aut}(N)$. Then $N \rtimes_{\phi}H\cong N\rtimes_{\psi\circ\phi}H$
So there will be at most one semidirect product. And we can assume that $\psi (c)=(12)$.
But isn't it the same as direct product? Is it true that if $\phi:H\to\mathrm{Inn}(N)$, than $N \rtimes_{\phi}H\cong N\times H$?