On a Combinatorial Characterization of Abelian Group Actions

Let $G$ be a finite abelian group of order $n$ and $X$ a finite $G$-set. We give a necessary and sufficient condition on $G$ for any $X$ to be determined up to isomorphism by certain combinatorial data. Specifically, if $M_H^X$ is the number of $H$-orbits in $X$, and $a_d^X$ is the number of $G$-orb