Extensions of tensor categories by finite group fusion categories

We study exact sequences of finite tensor categories of the form $\Rep G \to \C \to \D$,where$G$ is a finite group. We show that, under suitable assumptions, there exists a group $\mathrm{\Gamma }$ and mutual actions by permutations $⊳:\mathrm{\Gamma }\times G\to G$ and $⊲:\mathrm{\Gamma }\times G\to \mathrm{\Gamma }$ that make $(G,\mathrm{\Gamma })$ into matched pair of groups endowed with a natural crossed action on $\text{D}$ such that $\text{C}$ is equivalent to a certain associated crossed extension ${\text{D}}^{(G,\mathrm{\Gamma })}$ of $\text{D}$. Dually, we show that an exact sequence of finite tensor categories ${\text{vect}}_G\to \text{C}\to \text{D}$ induces an $\text{Aut}(G)$-grading on $\text{C}$ whose neutral homogeneous component is a $(Z(G),\mathrm{\Gamma })$-crossed extension of a tensor subcategory of $\text{D}$. As an application we prove that such extensions $\text{C}$ of $\text{D}$ are weakly group-theoretical fusion categories if and only if $\text{D}$ is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.