  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.
