Automorphism groups of nilpotent groups and spaces

A pair of finitely generated, torsion-free nilpotent groups G1,G2 is constructed with the properties that G1 and G2 are p-isomorphic for all primes p, yet Aut(G1) and Aut(G2) are not isomorphic. The example constructed is compared to an analogous example in the homotopy category of simply connected, finite CW-complexes.