  We give a classification of finite groups of symplectic birational
automorphisms on a manifold of K3^[3]-type with stable and stably saturated
cohomological action. We describe the group of polarized automorphisms of a
smooth double EPW-cube. Using this description, we exhibit examples of
projective hyperkaehler manifolds of K3^[3]-type of maximal Picard rank with a
symplectic action of a large group.
