  Let G be a connected algebraic group over an algebraically closed field of
characteristic p (possibly 0), and X a variety on which G acts transitively
with connected stabilizers. We show that any \'etale Galois cover of X of
degree prime to p is also homogeneous, and that the maximal prime-to-p quotient
of the \'etale fundamental group of X is commutative. We moreover obtain an
explicit bound for the number of topological generators of the said quotient.
When G is commutative, we also obtain a description of the prime-to-p torsion
in the Brauer group of G.
