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 $\omega\in H^3(G,\mathbb{ G}_m)$, 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 $\mathcal{ M}_{ \rm ext}(\Rep(\Gamma\times \alpha_p))\cong \g^{ *(1)}$.
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