  This is a sequel to the paper [Cas]. Here, we extend the methods of
Farb-Wolfson using the theory of FI_G-modules to obtain stability of
equivariant Galois representations of the etale cohomology of orbit
configuration spaces. We establish subexponential bounds on the growth of
unstable cohomology, and then use the Grothendieck-Lefschetz trace formula to
obtain results on arithmetic statistics for orbit configuration spaces over
finite fields. In particular, we show that the average value, across
polynomials over F_q, of certain Gauss sums over their roots, stabilizes as the
degree goes to infinity.
