Definability of completely decomposable torsion-free Abelian groups by semigroups of endomorphism and groups of homomorphisms

Let $C $ be an Abelian group. A class $X $ of Abelian groups is called a $_CE ^\bullet H $-class if for any groups $A,B \in X$, it follows from the existence of isomorphisms $E^\bullet (A) \cong E^\bullet (B)$ and $\operatorname{Hom}(C,A)\cong \operatorname{Hom}(C,B) $ that there is an isomorphism $A\cong B $. In this paper, conditions are studied under which the class $\Im _{\mathrm{cd}}^{\mathrm{ad}}$ of completely decomposable almost divisible Abelian groups and class $ \Im _{\mathrm{cd}}^{*} $ of completely decomposable torsion-free Abelian groups $A$ where $\Omega(A)$ contains only incomparable types are $_CE ^\bullet H $-classes, where $C $ is a completely decomposable torsion-free Abelian group.