K3 surfaces with a symplectic automorphism of order 4

Given $X$ a K3 surface admitting a symplectic automorphism $\tau$ of order 4, we describe the isometry $\tau^*$ on $H^2(X,\mathbb Z)$. Having called $\tilde Z$ and $\tilde Y$ respectively the minimal resolutions of the quotient surfaces $Z=X/\tau^2$ and $Y=X/\tau$, we also describe the maps induced in cohomology by the rational quotient maps $X\rightarrow\tilde Z,\ X\rightarrow\tilde Y$ and $\tilde Y\rightarrow\tilde Z$: with this knowledge, we are able to give a lattice-theoretic characterization of $\tilde Z$, and find the relation between the N\'eron-Severi lattices of $X,\tilde Z$ and $\tilde Y$ in the projective case. We also produce three different projective models for $X,\tilde Z$ and $\tilde Y$, each associated to a different polarization of degree 4 on $X$.