Geodesic
On Schur $2$-groups
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 28, Tome 435 (2015), pp. 113-162

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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. 
@article{ZNSL_2015_435_a6,
     author = {M. Muzychuk and I. Ponomarenko},
     title = {On {Schur} $2$-groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {113--162},
     publisher = {mathdoc},
     volume = {435},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_435_a6/}
}
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M. Muzychuk; I. Ponomarenko. On Schur $2$-groups. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 28, Tome 435 (2015), pp. 113-162. http://geodesic.mathdoc.fr/item/ZNSL_2015_435_a6/