  We study the properties of the fundamental group of an affine curve over an
algebraically closed field of characteristic $p$, from the point of view of
embedding problems. In characteristic zero, the fundamental group is free, but
in characteristic $p$ it is not even $\omega$-free. In this paper we show that
it is "almost $\omega$-free," in the sense that each finite embedding problem
has a proper solution when restricted to some open subgroup. We also prove that
embedding problems can always be properly solved over the given curve if
suitably many additional branch points are allowed, in locations that can be
specified arbitrarily; this strengthens a result of the first author.
