When is a group a semi-direct product with its normal subgroup?

Question

Let $G$ be an infinite non-abelian group. If we have a normal subgroup $N$ of $G$, then can we always construct the subgroup $H$ such that $G$ is a semidirect product of $N$ and $H$?

Answer 1

You cannot construct always that group, the necessary and sufficient condition for that is the fact that the exact sequence $1\rightarrow N\rightarrow G\rightarrow G/N\rightarrow 1$ splits

Answer 2

For a specific example, let $G = ${\mathbb Z}$ \times S_3$ (where ${\mathbb Z}$ denotes an infinite cyclic group), and $N = 2{\mathbb Z} \times S_3$. There is no such subgroup $H$.

Of course the standard example is $G = {\mathbb Z}$, $N = 2{\mathbb Z}$, but that is abelian, so I just took a direct product with $S_3$ to make it nonabelian.