On nilpotent Schur groups

A finite group $G$ is called a Schur group if every $S$-ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of $\mathrm{Sym}(G)$ that contains all right translations. We prove that every nonabelian nilpotent Schur group belongs to one of a few explicitly given families of groups.