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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 \(M=\mathcal{F}_TG\).

Some properties of E(G,W,F_TG) and an application in the theory of splittings of groups