Let $1 \to K \to \Gamma \to H \to 1$ be an exact sequence of groups. Show that $\Gamma$ is an amenable group if and only if $K, H$ are amenable.
I have already shown the first direction, by writing $H = \Gamma/K$ and playing with fundamental domains. However, I'm lost when I'm trying to prove the second direction (again, I think the fact that $H=\Gamma/K$ may be useful here). I have seen some proofs using invariant means, however I wanted to prove the statement in a more group theoretic way. The definition of amenability I have is that $\Gamma$ (countable) is amenable if there exists a finitely additive probability measure $\mu: 2^{\Gamma} \to [0,1]$ such that it is invariant under left translation.
What would be a way to get started with this? Any help would be appreciated!