As far as I know, a central + split extension of a group $K$ by a group $H$ is uniquely the direct product $H \times K$. But is the other direction true as well? That is, is any direct product necessarily a central and split extension?
I don't think so. Consider two non-abelian groups $H$ and $K$. And consider the direct product $H \times K$. A non-abelian group can never lie in the center of another group (because the center of any group is abelian). So neither $H$ nor $K$ can lie in the center of $H \times K$. But then $H \times K$ can't be a central extension of $H$ by $K$ or $K$ by $H$, isn't it? Could someone please clarify?