Last summer, while discussing possible formulations of a Riemann existence theorem for covers of the projective line in positive characteristic, I posed a question on the orbits of the braid group on certain tuples of permutations. This question may be formulated equivalently in terms of connected components of the Hurwitz spaces parametrizing certain branched covers of the projective line. I will discuss the answer to the question I posed, and explain how the result may be proved by a combination of group-theoretic and geometric techniques.