Kummer surfaces are special quartic surfaces that admit 16 nodes. The automorphisms of K3 Kummer surfaces are rich and complicated. Based on the results of Keum and Kondō, we lift 45 classically known automorphisms of a Kummer surface to pseudo-automorphisms of a threefold, the blow-up of P^3 along 6 very general points and 15 lines. We describe the subgroup generated by all the known pseudo-automorphisms of this threefold. This subgroup consists of all the pseudo-automorphisms that fix the unique rational normal curve through the 6 points.