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.
"Galois realization of Schur covers of dihedral groups of $2$-power order." Funct. Approx. Comment. Math. 67 (1) 47 - 67, September 2022. https://doi.org/10.7169/facm/1975