Embedding Obstructions for the Dihedral, Semidihedral, and Quaternion 2-Groups

For each of the dihedral, semidihedral, and quaternion 2-groups, we represent the obstructions to certain Brauer problems as tensor products of quaternion algebras. Then we reduce various embedding problems with cyclic 2-kernels into two Brauer problems, thus finding the obstructions in some specific cases.