Completely Decomposable Quotient Divisible Abelian Groups with Isomorphic Endomorphism Semigroups

Let $\Lambda$ be a class of Abelian groups. A group $A\in\Lambda$ is said to be determined by its endomorphism semigroup $E^\star(A)$ in the class $\Lambda$ if every isomorphism $E^\star(A)\cong E^\star(B)$, where $B\in\Lambda$, implies the isomorphism $A\cong B$. The paper describes those Abelian groups in the class $\mathscr Q\mathscr D_{\mathrm{cd}}$ of completely decomposable quotient divisible Abelian groups which are determined by their endomorphism semigroups in the class $\mathscr Q\mathscr D_{\mathrm{cd}}$.