Let G be a group. An automorphism of G is called an IA-automorphism if it induces the identity automorphism on \(G/G^{'}\) , where \(G^{'}\) is the derived subgroup. Denote by \(\textrm{IA}(G)^{*}\) the group of all IA-automorphisms of G fixing the centre elementwise. Suppose that G is a central product of a finitely generated nilpotent group and a nilpotent group with the minimal condition on subgroups. When the nilpotent class of G is 2, we prove that \(\textrm{IA}(G)^{*}\simeq \textrm{Inn}(G)\) if and only if \(G^{'}\) is cyclic.
