Finite torsors over strongly $F$-regular singularities

We investigate finite torsors over big opens of spectra strongly $F$-regular germs that do not extend to the whole spectrum. Let $(R,\mathfrak{m},k)$ be a $k$-germ where $k$ is an algebraically closed field characteristic $p>0$. prove existence local cover $R \subset R^{\star}$ so $R^{\star}$ and: for all algebraic groups $G/k$ with solvable neutral component, every $G$-torsor open $\mathrm{Spec} extends everywhere. To achieve this, we obtain generalized transformation rule $F$-signature under extensions. Such formula used show torsion $\mathrm{Cl} R$ bounded by $1/s(R)$. By taking cones, conclude Picard group globally varieties torsion-free. Likewise, it shows canonical covers $\mathbb{Q}$-Gorenstein singularities are $F$-regular.