These are the (torsion) cotorsion groups. The following follows from a theorem of Baer:
A torsion abelian group is cotorsion if and only it is direct sum of a divisible torsion abelian group and an abelian group of bounded exponent.
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. (See in particular Theorem 8.1.)
[I have made Baer's paper available here. --PLC]
