Let $G$ be a finite group. By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$. One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis permuted by $G$.
Question. For a $G$-lattice $L$, how can one compute the minimal $\mathbb{Z}$-rank of a permutation $G$-lattice $P$ such that there exists a surjective $G$-morphism $P\to L$ ?
I expect an answer in terms of some kind of group cohomology. I would be happy to get an answer at least in the cases when $G$ is a $p$-group, an abelian $p$-group, a cyclic $p$-group.
