Central extension and direct product

Question

Let $G,H,K$ be finite groups and suppose $G$ is a central extension of $H$ by $K$. The question is that under which condition on this extension we will have $G \cong K \times H$.

Answer 1

We will have $G \cong K \times H$, if and only if the extension $$ 1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1 $$ splits, which means that its equivalence class is the trivial one in the second cohomology group $H^2(H,K)$.

Answer 2

I am not sure what sort of answer you are looking for, but it is true that it is a direct sum if and only if the sequence left splits, i.e. if for the short exact sequence $1 \rightarrow K \xrightarrow{\psi} G \rightarrow H \rightarrow 1$ there is a group homomorphism $\phi: G \rightarrow K$ "back" in such a way that $\phi \circ \psi = id_K $. This is part of the Splitting Lemma.