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

Motivic classes of generalized Kummer schemes via relative power structures