Mapping near-rings of abelian groups

Let G be an abelian group, and let p be a prime. A mapping f : G → G is said to be p-homogeneous if f(px) = pf(x) for all x ∈ G. If every p-homogeneous mapping of G is an endomorphism, then G is said to be p-endomorphal. If the set M_p(G), consisting of all p-homogeneous maps of G, forms a ring under the operations point-wise addition and composition, then G is said to be semi-p-endomorphal. It is shown that G is p-endomorphal if and only if G is semi-p-endomorphal. The p-endomorphal groups are described completely.