Isomorphisms between centers of integral group rings

For finite nilpotent groups $G$ and $G^{\prime}$, 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 $SG^{\prime}$ is monomial, i.e., maps class sums in $SG$ to class sums in $SG^{\prime}$ up to multiplication with roots of unity. As a consequence, $G$ and $G^{\prime}$ have identical character tables if and only if the centers of their integral group rings $\mathbb{Z} G$ and $\mathbb{Z} G^{\prime}$ are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.