on the properties of semidirect product of finite groups

Question

If $G$ is a finite group such that $G/H \cong K$, where $H,K \leq G$, then is it true that $G \cong H \rtimes K$? it is sufficient to show that $H \cap K=1$ and $G=HK$.

Answer 1

No, it isn't. For example with the quaternion group

$$\mathcal Q_8=\{1,-1,i,j,k,-i,-j,-k\}\;,\;\;\mathcal Q_8/\langle i\rangle\cong\{1,-1\}\require{cancel} \cancel{\implies}\mathcal Q_8\cong \langle i\rangle\rtimes\langle -1\rangle$$

because, of course, $\;\langle -1\rangle\subset\langle i\rangle\;$ ... In fact, the subgroup of order two in this case is unique (and thus normal), and it is contained in any other non-trivial subgroup of $\;\mathcal Q_8\;$ .