Kummer's quartics and numerically reflective involutions of Enriques surfaces

A (holomorphic) involution sigma of an Enriques surface S is said to be numerically reflective if it acts on the cohomology group H-2(S, Q) as a reflection. We show that the invariant sublattice H(S, sigma; Z) of the anti-Enriques lattice H- (S, Z) under the action of sigma is isomorphic to either (-4) perpendicular to U(2) perpendicular to U(2) or <-4 > perpendicular to U(2) perpendicular to U. Moreover, when H(S, sigma; Z) is isomorphic to <-4 > perpendicular to U(2) perpendicular to U(2), we describe (S, sigma) geometrically in terms of a curve of genus two and a Gopel subgroup of its Jacobian.