The classical Kummer construction associates a K3 surface to any 2-dimensional complex torus. In my talk I will present an analogue of this construction, which involves the two most well-studied deformation types of hyper-Kähler manifolds in dimension 6. Namely, starting from any hyper-Kähler sixfold K of generalized Kummer type, I am able to construct geometrically a hyper-Kähler manifold of K3^[3]-type. When K is projective, the associated variety is birational to a moduli space of sheaves on a uniquely determined K3 surface. As application of this construction I will show that the Kuga-Satake correspondence is algebraic for many K3 surfaces of Picard rank 16.