Galois extensions, plus closure, and maps on local cohomology

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 is zero for each . We prove that the extension S may be chosen to be generically Galois, and analyze the Galois groups that arise.