Relative group (co)homology theories with coefficients and the comparison homomorphism

Let $G$ be a group, let $H$ be a subgroup of $G$ and let $\text{Or}(G)$ be the orbit category. In this paper we extend the definition of the relative group (co)homology theories of the pair $(G,H)$ defined by Adamson and Takasu to have coefficients in an $\text{Or}(G)$-module. There is a canonical comparison homomorphism defined by Cisneros-Molina and Arciniega-Nev'arez from Takasu’s theory to Adamson’s one. We give a necessary and sufficient condition on the subgroup $H$ for which the comparison homomorphism is an isomorphism for all coefficients. We also use the L"uck-Wiermann construction to introduce a long exact sequence for Adamson (co)homology. Finally, we provide some examples of explicit computations for the comparison homomorphism.