On extensions of bounded subgroups in Abelian groups

It is well-known that every bounded Abelian group is a direct sum of finite cyclic subgroups. We characterize those non-trivial bounded subgroups H of an infinite Abelian group G, for which there is an infinite subgroup G₀ of G containing H such that G₀ has a special decomposition into a direct sum which takes into account the properties of G, and which induces a natural decomposition of H into a direct sum of finite subgroups.