Abelian groups as endomorphic modules over their endomorphism ring

Let $R$ be an associative ring with a unit and $N$ be a left $R$-module. The set $M_R(N)=\{f\colon N\to N\mid f(rx)=rf(x),\ r\in R,\ x\in N\}$ is a near-ring with respect to the operations of addition and composition and contains the ring $E_R(N)$ of all endomorphisms of the $R$-module $N$. The $R$-module $N$ is endomorphic if $M_R(N)=E_R(N)$. We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.