If $K\le N$, what are the subgroups $\Gamma\le N\rtimes H$ such that $\Gamma\cap N = K$ and $\Gamma$ surjects onto $H$?

Question

Let $N$ be a group, and $K\le N$ a finite index normal subgroup. Consider a semi-direct product $N\rtimes H$. (I'm happy to assume all groups are finitely generated residually finite profinite groups)

I would like to understand the subgroups $\Gamma\le N\rtimes H$ such that $\Gamma\cap N = K$, and $\Gamma$ surjects onto $H$.

1. If $K\le N$ is invariant under the action of $H$, then we can form $\Gamma := K\rtimes H$, which certainly satisfies our conditions. Can there exist other possibilities for $\Gamma$? Can we "classify" them? (perhaps via some kind of cohomology?)

2. If a $\Gamma$ exists, must $K$ be invariant under the action of $H$ on $N$?

3. If (2) is false, can we describe the possibilities for $\Gamma$ when $K$ is not invariant?