  We develop a power structure over the Grothendieck ring of varieties relative
to an abelian monoid, which allows us to compute the motivic class of the
generalized Kummer scheme. We obtain a generalized version of Cheah's formula
for the Hilbert scheme of points, which specializes to Gulbrandsen's conjecture
for Euler characteristics. Moreover, in the surface case we prove a conjecture
of G\"{o}ttsche for geometrically ruled surfaces, and we obtain an explicit
formula for the virtual motive of the generalized Kummer scheme in dimension
three.
