Maps between spherical group rings

We prove that for finitely generated abelian groups $A$ and $B$, the space of ${\mathbb{E}}_{\mathrm{\infty }}$-ring maps between the spherical groups rings $\mathbb{S}[A]\to \mathbb{S}[B]$ is equivalent to the discrete set of group homomorphisms $A\to B$. We also prove generalizations where the sphere is replaced by other ring spectra, e.g. we give a formula for the strict units in group rings of the form $R[A]$ for $A$ a finite $p$-group and $R$ $p$-completely chromatically complete.