K3 surfaces with a symplectic automorphism of order 4

Given (Formula presented.), a K3 surface admitting a symplectic automorphism (Formula presented.) of order 4, we describe the isometry (Formula presented.) on (Formula presented.). Having called (Formula presented.) and (Formula presented.), respectively, the minimal resolutions of the quotient surfaces (Formula presented.) and (Formula presented.), we also describe the maps induced in cohomology by the rational quotient maps (Formula presented.) and (Formula presented.) : With this knowledge, we are able to give a lattice-theoretic characterization of (Formula presented.), and find the relation between the Néron–Severi lattices of (Formula presented.) and (Formula presented.) in the projective case. We also produce three different projective models for (Formula presented.) and (Formula presented.), each associated to a different polarization of degree 4 on (Formula presented.).