  Let $G$ be a finite group and $N<G$ a normal subgroup with $G/N$ abelian. We
show how the conjugacy classes of $G$ in a given coset $qN$ relate to the
irreducible characters of $G$ that are not identically $0$ on $qN$. We describe
several consequences. In particular, we deduce that when $G/N$ is cyclic
generated by $q$, the number of irreducible characters of $N$ that extend to
$G$ is the number of conjugacy classes of $G$ in $qN$.
