Outer automorphism groups of metabelian groups

Recall that the outer automorphism group of a group G, denoted Out G, is the quotient group AutG/Inn G. If M is any group, then there exists a torsion-free, metabelian group G with trivial center such that Out G M. This answers a problem in the Kourovka Notebook (Mazurov, Khukhro, Unsolved problems in group theory; the Kourovka Notebook, Russian Academy of Science, Novosibirsk, 1992).