On isomorphisms between centers of integral group rings of finite groups

For finite nilpotent groups G and G', and a G-adapted ring S ( the rational integers, for example), it is shown that any isomorphism between the centers of the group rings SG and SG' is monomial, i.e., maps class sums in SG to class sums in SG' up to multiplication with roots of unity. As a consequence, G and G' have identical character tables if and only if the centers of their integral group rings ZG and ZG' are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.