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, m, k, 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 of R* so that R* 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 SpecR* 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 the torsion of C1 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, this shows that canonical covers of Q-Gorenstein strongly F-regular singularities are strongly F-regular.