  We analyse the geometry of Hilbert schemes of points on abelian surfaces and
Beauville's generalized Kummer varieties in positive characteristics. The main
result is that, in characteristic two, the addition map from the Hilbert scheme
of two points to the abelian surface is a quasifibration, such that all fibers
are nonsmooth. In particular, the corresponding generalized Kummer surface is
nonsmooth, and minimally elliptic singularities occur in the supersingular
case. We unravel the structure of the singularities in dependence of p-rank and
a-number of the abelian surface. To do so, we establish a McKay Correspondence
for Artin's wild involutions.
