Some properties of $E(G,W,\mathcal{F}_TG)$ and an application in the theory of splittings of groups

Let us consider $W$ a $G$-set and $M$ a $\mathbb{Z}_2G$-module, where $G$ is a group. In this paper we investigate some properties of the cohomological the theory of splittings of groups. Namely, we give a proof of the invariant $E(G,W,M)$, defined in [5] and present related results with independence of $E(G,W,M)$ with respect to the set of $G$-orbit representatives in $W$ and properties of the invariant $E(G,W,\mathcal{F}_TG)$ establishing a relation with the end of pairs of groups $\widetilde{e}(G,T)$, defined by Kropphller and Holler in [15]. The main results give necessary conditions for $G$ to split over a subgroup $T$, in the cases where $M=\mathbb{Z}_2(G/T)$ or $M=\mathcal{F}_TG$.