On Steinitz classes, realizable Galois module classes and embedding problems for non-abelian extensions of degree a power of 2

Let Gamma be the non-abelian group of order 16 which is the central product of the dihedral group of order 8 and the cyclic group of order 4, or an extra-special group of order 32. In the present article, we study an embedding problem connected with Steinitz classes in the perspective of studying the structure of the set of Galois module classes realizable by the rings of integers of extensions with Galois group isomorphic to Gamma.