  Let $G$ be a group, let $H$ be a subgroup of $G$ and let $\Or(G)$ 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 $\Or(G)$-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.
