Let \({\mathfrak X}\) denote the class of all finitely generated torsion-free nilpotent groups G such that the derived factor group G/G' is torsion- free. For G in \({\mathfrak X}\), let Aut *(G) denote the group of automorphisms of G/G' induced by the automorphism group of G. If G/G' has rank n and we choose a \({\mathbb{Z}}\)-basis for G/G' then Aut *(G) can be regarded as a subgroup of GL(n,\({\mathbb{Z}})\). The first author and \textit{J. R. J. Groves} [J. Lond. Math. Soc., II. Ser. 33, 453-466 (1986; Zbl 0554.20008)] showed that every arithmetic group is commensurable with Aut *(G) for some G in \({\mathfrak X}\). The present paper contains a sharper result. If A is any Zariski-closed subgroup of GL(n,\({\mathbb{Z}})\), where \(n\geq 2\), then there exists G in \({\mathfrak X}\) and a basis for G/G' such that \(A=Aut\) *(G). It follows that if S is any finitely generated nilpotent-by-finite group then there exists G in \({\mathfrak X}\) such that Aut *(G) is isomorphic to S.