Kummer sandwiches and Greene–Plesser construction

In the context of K3 mirror symmetry, the Greene–Plesser orbifolding method constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in P3, starting from a special one-parameter family of K3 varieties known as the quartic Dwork pencil. We show that certain K3 double covers obtained from the three-parameter family of quartic Kummer surfaces associated with a principally polarized abelian surface generalize the relation of the Dwork pencil and the quartic mirror family. Moreover, for the three-parameter family we compute a formula for the rational point-count of its generic member and derive its transformation behavior with respect to (2,2)-isogenies of the underlying abelian surface.