  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
$\mathcal{M}_{\rm ext}(\Rep(\Gamma\times \alpha_p))\cong \g^{*(1)}$.
