  We prove that the local etale fundamental group of a strongly $F$-regular
singularity is finite (and likewise for the \'etale fundamental group of the
complement of a codimension $\geq 2$ set), analogous to results of Xu and
Greb-Kebekus-Peternell for KLT singularities in characteristic zero. In fact
our result is effective, we show that the reciprocal of the $F$-signature of
the singularity gives a bound on the size of this fundamental group. To prove
these results and their corollaries, we develop new transformation rules for
the $F$-signature under finite etale-in-codimension-one extensions. As another
consequence of these transformation rules, we also obtain purity of the branch
locus over rings with mild singularities (particularly if the $F$-signature is
$> 1/2$). Finally, we generalize our $F$-signature transformation rules to the
context of pairs and not-necessarily etale-in-codimension-one extensions,
obtaining an analog of another result of Xu.
