K3 surfaces with a symplectic action of (Z/2 Z) ²

We study the symplectic action of the group (Z/2Z) ² on a K3 surface X: we describe its action on H² (X, Z) and the maps induced in cohomology by the rational quotient maps; we give a lattice-theoretic characterization of the resolution of singularities of the quotient X/i, where i is any of the involutions in (Z/2Z) ². Assuming X is projective, we describe the correspondence between irreducible components of its moduli space, and those of the resolution of singularities of its quotients: this being the first description of this correspondence for a non-cyclic action, we see new phenomena, of which we provide explicit examples assuming X has a polarization of degree 4.