On the central extension of a group

Question

Let $G$ be a central extension of a group $K$ by the perfect group $H$ ($K$ is the normal subgroup).

The question is that if $(|K|,|Mult(H)|)=1$ then can we say that $G=H \times K$?

I would be grateful if anyone could answer the question.

Answer 1

I guess ${\rm Mult}(H)$ means the Schur Multiplier of $H$.

Assuming that all the groups are finite, the answer is yes, because $G' \cap K$ is isomorphic to a subgroup of ${\rm Mult}(H)$ and hence $G ' \cap K = 1$.