On a class of quotient divisible Abelian groups with isomorphic endomorphism semigroups

Let $\Lambda$ be some class of Abelian groups. We say that a group $A\in\Lambda $ is determined by its endomorphism semigroup $E^\star(A)$ in $\Lambda $ if the isomorphism $E^\star(A)\cong E^\star(B)$, where $B\in\Lambda$, implies $A\cong B$. We describe split Abelian groups in the class $\mathcal{QD}_{\mathrm{cd}}$ of completely decomposable quotient divisible Abelian groups determined by their endomorphism semigroups in the class $\mathcal{QD}_{\mathrm{cd}}$.