  We employ methods from homotopy theory to define new obstructions to
solutions of embedding problems. By using these novel obstructions we study
embedding problems with non-solvable kernel. We apply these obstructions to
study the unramified inverse Galois problem. That is, we show that our methods
can be used to determine that certain groups cannot be realized as the Galois
groups of unramified extensions of certain number fields. To demonstrate the
power of our methods, we give an infinite family of totally imaginary quadratic
number fields such that $\text{Aut}(\text{PSL}(2,q^2))$ for $q$ an odd prime
power, cannot be realized as an unramified Galois group over $K,$ but its
maximal solvable quotient can. To prove this result, we determine the ring
structure of the \'etale cohomology ring $H^*(\text{Spec
}\mathcal{O}_K;\mathbb{Z}/ 2\mathbb{Z})$ where $\mathcal{O}_K$ is the ring of
integers of an arbitrary totally imaginary number field $K.$
