  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 $X$.
