On definability of a periodic $\mathrm{EndE}^+$-group by its endomorphism group

Let $\mathbf A$ be a class of Abelian groups, $A\in\mathbf A$, and $\mathrm{End}(A)$ be the additive endomorphism group of the group $A$. The group $A$ is said to be defined by its endomorphism group in the class $\mathbf B\supseteq\mathbf A$ if for every group $B\in \mathbf B$ such that $\mathrm{End}(B)\cong\mathrm{End}(A)$ the isomorphism $B\cong A$ holds. The paper considers the problem of definability of a periodic Abelian group $A$ such that $\mathrm{End}\bigl(\mathrm{End}(A)\bigr)\cong\mathrm{End}(A)$. The classes of periodical Abelian groups, of divisible Abelian groups, of reduced Abelian groups, of nonreduced Abelian groups, and of all Abelian groups are investigated in this paper.