Given a cotriple 𝔾 = (G, ε, δ) on a categoryXand a functor E:XOpp→Ainto an abelian categoryA, there exists the cohomology theory of Barr and Beck: Hn(X, E) ε |A| (n ≥ 0, X ε |X|), ([1], p.249). Almost all the important cohomology theories in mathematics have been shown to be special instances of such a general theory (see [1], [2] and [3]). Usually E arises from an abelian group object Y inXin the following manner: it is the contravariant functor fromXinto the categoryAbof abelian groups that associates to each object X inXthe abelian groupX(X, Y) of maps from X to Y. In such a situation we shall write Hn(X, Y)𝔾instead of Hn(X, E)G. Barr and Beck [2] have shown that the Eilenberg-MacLane cohomology groups H̄n(π, A), n ≥ 2, can be re-captured as follows. One considers the free group cotriple 𝔾′ on the categoryGpsof groups, which induces in a natural manner a cotriple 𝔾 on the category (Gps, π) of groups over a fixed group π.