  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.
