  The classical Kummer construction attaches to an abelian surface a K3
surface. As Shioda and Katsura showed, this construction breaks down for
supersingular abelian surfaces in characteristic two. Replacing supersingular
abelian surfaces by the selfproduct of the rational cuspidal curve, and the
sign involution by suitable infinitesimal group scheme actions, I give the
correct Kummer-type construction in this situation. We encounter rational
double points of type D_4 and D_8, instead of type A_1. It turns out that the
resulting surfaces are supersingular K3 surfaces with Artin invariant one and
two. They lie in a 1-dimensional family obtained by simultaneous resolution
after purely inseparable base change.
