  We study the symplectic action of the group (Z/2Z)^2 on a K3 surface X: we
describe its action on H^2(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.
