Galois realizability of non-split group extensions of C2 by (C2)r×(C4)s×(D4)t

The focus of this paper is Galois embedding problems associated with extensions of C2 by groups of the form (C2)r×(C4)s×(D4)t over arbitrary fields of characteristic not 2. As an application of our general solution, we obtain obstructions to the realizability of 13 groups of order 32 for which obstructions were not previously known.