Induced orthogonal representations of Galois groups

We prove a result showing the connection between induced orthogonal representations and corestrictions of group extensions derived from their Clifford groups. By the means of the corestriction map we then obtain new obstructions to the μp-embedding problems given by the group extensions of the modular p-group, and to one μ2-embedding problem given by a group extension of the dihedral group.