Character formula for conjugacy classes in a coset

Abstract: 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 $qN$.