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