Galois realization of Schur covers of dihedral groups of $2$-power order

Abstract

The Schur covers of a dihedral group of $2$-power order are a dihedral group,a semi-dihedral group, and a generalized quaternion group.We prove that if one of the three groups is realizable as Galois group, thenother two groups are also realizable as Galois groups under a~condition on the existence of a certain quartic cyclic extension over the base field.