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We show that any slightly degenerate weakly group-theoretical fusion categoryadmits a minimal non-degenerate extension. Let $d$ be a positive square-freeinteger, given a weakly group-theoretical non-degenerate fusion category$\mathcal{C}$, assume that $\text{FPdim}(\mathcal{C})=nd$ and $(n,d)=1$. If$(\text{FPdim}(X)^2,d)=1$ for all simple objects $X$ of $\mathcal{C}$, then weshow that $\mathcal{C}$ contains a non-degenerate fusion subcategory$\mathcal{C}(\mathbb{Z}_d,q)$. In particular, we obtain that integral fusioncategories of FP-dimensions $p^md$ such that $\mathcal{C}'\subseteq\text{sVec}$ are nilpotent and group-theoretical, where $p$ is a prime and$(p,d)=1$.

On the minimal extension and structure of weakly group-theoretical braided fusion categories