Let \(S\) be a monoid and \(A\) a right \(S\)-set. A right \(S\)-set over \(A\) is a pair of a right \(S\)-set \(X\) and a homomorphism of \(S\)-sets \(\xi\colon X\to A\); \(\overline{\mathbf C}\) is the category of all right \(S\)-sets over \(A\) and their homomorphisms. A right \(S\)-set \(\overline G=(G,\gamma)\) over \(A\) together with an Abelian group operation \(+\) on every \(\Hom_{\overline{\mathbf C}}(\overline X,\overline G)\) with the property \((g+h)\circ f=(g\circ f)+(h\circ f)\) is called an Abelian group object of \(\overline{\mathbf C}\). It is shown that Abelian group objects which are coefficients in the triple cohomology of \(A\) can be presented as Abelian group valued functors on \(A\) and Beck extensions of an Abelian group valued functor \(G\) by \(A\) may be identified with group coextensions of \(A\) by \(G\); these results are used to describe basic properties of the triple cohomology of right \(S\)-sets.