  Given a local domain $(R,m)$ of prime characteristic that is a homomorphic
image of a Gorenstein ring, Huneke and Lyubeznik proved that there exists a
module-finite extension domain $S$ such that the induced map on local
cohomology modules $H^i_m(R)\to H^i_m(S)$ is zero for each $i<\dim R$. We prove
that the extension $S$ may be chosen to be generically Galois, and analyze the
Galois groups that arise.
