Kummer surfaces associated with group schemes

We introduce Kummer surfaces X=Km(CxC) with the group scheme G=mu_2 acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational double points of type A_1, together with a rational double point of type D_4. We show that our Kummer surfaces are precisely the supersingular K3 surfaces with Artin invariant sigma\leq 3, and characterize them by the existence of a certain configuration of thirty curves. After contracting suitable curves, they also appear as normal K3-like coverings for simply-connected Enriques surfaces.