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In this paper, we consider for any free presentation G=F/R of a group G the coinvariance H₀(G,R_ab^⊗n) of the nth tensor power of the relation module R_ab and show that the homology group H_2n(G,Z) may be identified with the limit of the groups H₀(G,R_ab^⊗n), where the limit is taken over the category of these presentations of G. We also consider the free Lie ring generated by the relation module R_ab, in order to relate the limit of the groups γ_nR/[γ_nR,F] to the n-torsion subgroup of H_2n(G,Z).