I have a group extension $1\rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ that I think is a split extension (so $G \approx N \rtimes Q$), but I'm having trouble showing this. Is there a general strategy for how to define the section $Q \rightarrow G$? If possible, I would like to have an explicit mapping $Q \rightarrow G$ or $N \rtimes Q \overset{\sim}{\rightarrow} G$.
For some additional details on the specific case, $G \leq H \wr_{\Omega} Q = \mathrm{Fun(\Omega, H)} \rtimes Q$ and $N = \ker (G \twoheadrightarrow Q)$ (where this mapping is surjective).
EDIT: I've computed some examples of $G$ (which show up as monodromy groups of certain coverings) and in each case, it seemed to be a semidirect product: $C_{m} \rtimes C_{2}$ (for any $m$), $A_{7} \wr C_{2}$, $(S_{5} \times S_{5}) \rtimes C_{2}$, $C_{7} \rtimes A_{5}$, $(C_{2}^{18} \rtimes C_{3}^{8}) \rtimes S_{9}$, $C_{7}^{6} \rtimes A_{5}$, etc.
However, when I was going back through all of my examples, I found $C_{2}^{6} \rtimes C_{9}$, but as an extension $1 \rightarrow C_{2}^{6} \rtimes C_{3} \rightarrow C_{2}^{6} \rtimes C_{9} \rightarrow C_{3} \rightarrow 1$. Based on what I've read about $C_{4}$ not being a split extension of $C_{3}$, I assume that $C_{2}^{6} \rtimes C_{9} \not\approx (C_{2}^{6} \rtimes C_{3}) \rtimes C_{3}$ and $G$ is, in fact, not always a split extension.
This might be why I was having trouble defining the section $Q \rightarrow G$ in the general case.