Given a group G, I use the terminology G->G or better M(G) for the mappings from G to itself. Some of these mappings are special, in particular those mappings that respect the addition on the group G. A mapping f from M(G) respects the addition if f(a+b) = f(a) + f(b) for all a,b in G and is called an endomorphism. We will use the terminology E(G) for the set of all endomorphisms on a group.
Given two mappings f,g from M(G), we can define their sum as a point-wise sum; that is (f+g)(x) = g(x) + g(x). This is the usual sum of functions that we use. The other operation of interest is composition. The composition of two functions f,g from M(G) is (fog)(x) = f(g(x)).
Thus we come up with two operations on M(G). If we have an abelian group, then the set E(G) is closed under these two operations. That is, the sum of two endomorphisms is an endomorphism, and the composition of two endomorphisms is an endomorphism. Thus the algebra (E(G), +, o) is a ring.
Similarly, if we take take a non abelian group, or in fact any group at all, we can look at the collection of all mappings on that group under pointwise addition and composition and find that we have a near-ring (M(G),+,o). This will be a right near-ring since ((f+g)oh)(x) = (f+g)(h(x)) = (f(h(x)) + g(h(x))) = (foh + goh)(x). In general, the set of endomorphisms of a non-abelian group is not closed. Thus we speak of the set of mappings generated by the endomorphisms under the operations of pointwise addition and composition, and still use the terminology E(G) for this near-ring (E(G),+,o).