  In the context of K3 mirror symmetry, the Greene-Plesser orbifolding method
constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in
$\mathbb{P}^3$, 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.
