These are the (torsion) cotorsion groups. By a theorem of Baer all torsion abelian groups that are cotorsion are direct sums of a divisible group and a group of bounded order.
The original paper of R. Baer is "The subgroup of the elements of finite order of an abelian group", Ann. of Math. 37 (1936), 766-781.
[I have made Baer's paper available here. --PLC]
