Central extension Let $K$ be a cyclic group of order 12 and $Q$ be projective special linear group $PSL(4,4)$. If $G$ is a central extension of $K$ by $Q$ ($K$ is normal in $G$), then how many choices we have for the group $G$? Actually I wonder whether $G$ can be a non-split extension of $K$ by $Q$?

Apart from a few exceptional cases ($(n,q)=(2,4),(2,9),(3,2),(3,4),(4,2)$), the Schur Multiplier of the group ${\rm PSL}(n,q)$ is cyclic of order $\gcd(n,q-1)$.
So ${\rm PSL}(4,4)$ has trivial Schur Multiplier, and the only central extension of the type you describe is the direct product.
