Inertial automorphisms of an abelian group

An automorphisms $\gamma$ of a group is inertial if $X\cap X^\g$ has finite index in both $X$ and $X^\g$ for any subgroup $X$. We study inertial automorphisms of abelian groups and give characterization of them. In particular, if the group is periodic they have property that $X^{}/X_{}$ is bounded. We also study finitely generated groups of inertial automorphisms