Endomorphism of Abelian Groups as Modules over Their Endomorphism Rings

For an Abelian group $A$, viewed as a module over its endomorphism ring $E(A)$, the near-ring $\mathcal{M}_{E(A)}(A)$ of homogeneous mappings is defined as the set of mappings $\{f\colon A\to A \mid f(\varphi a)=\varphi f(a)$ for all $\varphi\in E(A)$ and $a\in A\}$ with the operations of addition and composition (as multiplication). It is proved that the problem of describing some classes of mixed Abelian groups with the property $\mathcal{M}_{E(A)}(A)=E(A)$ reduces to the cause of torsion-free Abelian groups. Abelian groups with this property are found in the class of strongly indecomposable torsion-free Abelian groups of finite rank and torsion-free Abelian groups of finite rank coinciding with their pseudosocle.