Galois extensions,plus closure,and maps on local cohomology

Given a local domain $(R,\mathfrak{m})$ 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 $H_{\mathfrak{m}}^i(R)\to H_{\mathfrak{m}}^i(S)$ 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.