On Schur $2$-groups

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a point stabilizer in a subgroup of $\operatorname{Sym}(G)$ that contains all right translations. We complete a classification of abelian Schur $2$-groups by proving that the group $\mathbb Z_2\times\mathbb Z_{2^n}$ is Schur. We also prove that any non-abelian Schur $2$-group of order larger than $32$ is dihedral (the Schur $2$-groups of smaller orders are known). Finally, in the dihedral case, we study Schur rings of rank at most $5$, and show that the unique obstacle here is a hypothetical S-ring of rank $5$ associated with a divisible difference set.