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

The unramified inverse Galois problem and cohomology rings of totally imaginary number fields