Let $G$ be a finite group. Is it true that every subgroup of $G\rtimes_{\varphi} \mathbb{Z}/2\mathbb{Z}$ is of the form $H$ or $H\rtimes_{\varphi} \mathbb{Z}/2\mathbb{Z}$, where $H\subset G$ is a subgroup?
It is known that, in general, the structure of subgroups of a semidirect product is somewhat tricky. What can we say in this simple case?
No. Take the semidirect (in fact, just direct) product $\;\Bbb Z_2\times\Bbb Z_2\;$ , which is the Klein group (the non-cyclic group of order four). Then, the subgroup $\;K:=\langle\,(1,1)\,\rangle\;$ is neither $\;K\le\Bbb Z_2\;$ nor $\;K\rtimes\Bbb Z_2\;$