Definability of Completely Decomposable Torsion-Free Abelian Groups by Endomorphism Semigroups and Homomorphism Groups

Let $C$ be an Abelian group. A class $X$ of Abelian groups is called a ${}_CE^\bullet H$-class if, for every groups $A,B\in X$, the isomorphisms $E^\bullet(A)\cong E^\bullet(B)$ and $\operatorname{Hom}(C,A)\cong\operatorname{Hom}(C,B)$ imply the isomorphism $A\cong B$.
In the paper, necessary and sufficient conditions on a completely decomposable torsion-free Abelian group $C$ are described under which a given class of torsion-free Abelian groups is a ${}_CE^\bullet H$-class.