Embedding Obstructions for the Generalized Quaternion Group

In this paper, we examine the obstructions to the solvability of certain embedding problems with the generalized quaternion group over arbitrary fields of characteristic not 2. First we consider the Galois embedding problem with abelian kernel in cohomological terms. Then we proceed with a number of examples in order to illustrate the role of the properties of the base field on the solvability of the embedding problem.