Abelian groups with finitely approximated acts

The Abelian groups of each of the following classes are completely described: ($*$) class of Abelian groups such that all the acts over them are finitely approximated, ($**$) Abelian groups such that all the acts over them are approximated by the acts consisting of $n$ or less elements, where $n$ is some natural number. In particular, a group belongs to the class ($**$) if and only if it is bounded.