We prove that any hyper-Kähler sixfold K of generalized Kummer type has a naturally associated manifold YK of K3[3] type. It is obtained as crepant resolution of the quotient of K by a group of symplectic involutions acting trivially on its second cohomology. When K is projective, the variety YK is birational to a moduli space of stable sheaves on a uniquely determined projective K3 surface SK. As an application of this construction we show that the Kuga–Satake correspondence is algebraic for the K3 surfaces SK, producing infinitely many new families of K3 surfaces of general Picard rank 16 satisfying the Kuga–Satake Hodge conjecture.