When is an Abelian group isomorphic to its endomorphism group?

In the paper, necessary and sufficient conditions for an Abelian group $A$ to be isomorphic to the endomorphism group $\operatorname{End}(A)$ are obtained. The classes of periodic Abelian groups, divisible Abelian groups, nonreduced Abelian groups, and reduced algebraically compact Abelian groups are considered. For certain classes of Abelian groups, the isomorphism problem for a group and its endomorphism group is solved under the assumption that the endomorphism group itself has the corresponding property.