Let $A$ be a finite abelian group, $k$ an algebraically closed field of characteristic $0$ and $\phi$ a $2$-cocycle on $A$ with value in $k$.
Do we know all the representations of the central extension $E$ of $A$ by $k$ given by $\phi$? If so, by what are they clasified?
Also, which representations of $E$ are trivial on $k$?