Minimal extensions of Tannakian categories in positive characteristic

We extend cite[Theorem 4.5]{DGNO} and cite[Theorem 4.22]{LKW} to positive characteristic (i.e., to the finite, not necessarily fusion, case). Namely, we prove that if $D$ is a finite non-degenerate braided tensor category over an algebraically closed field $k$ of characteristic $p>0$, containing a Tannakian Lagrangian subcategory $Rep(G)$, where $G$ is a finite $k$-group scheme, then $D$ is braided tensor equivalent to $Rep(D^{omega}(G))$ for some $omegain H^3(G,mathbb{G}_m)$, where $D^{omega}(G)$ denotes the twisted double of $G$ cite{G2}. We then prove that the group $mathcal{M}_{{rm ext}}(Rep(G))$ of minimal extensions of $Rep(G)$ is isomorphic to the group $H^3(G,mathbb{G}_m)$. In particular, we use cite{EG2,FP} to show that $mathcal{M}_{rm ext}(Rep(mu_p))=1$, $mathcal{M}_{rm ext}(Rep(alpha_p))$ is infinite, and if $O(Gamma)^*=u(g)$ for a semisimple restricted $p$-Lie algebra $g$, then $mathcal{M}_{rm ext}(Rep(Gamma))=1$ and $mathcal{M}_{rm ext}(Rep(Gammatimes alpha_p))cong g^{*(1)}$.