The group of inertial automorphisms of an abelian group

We study the group $IAut(A)$ generated by the inertial automorphisms of an abelian group $A$, that is, automorphisms $γ$ with the property that each subgroup $H$ of $A$ has finite index in the subgroup generated by $H$ and $Hγ$. Clearly, $IAut(A)$ contains the group $FAut(A)$ of finitary automorphisms of $A$, which is known to be locally finite. In a previous paper, we showed that $IAut(A)$ is (locally finite)-by-abelian. In this paper, we show that $IAut(A)$ is also metabelian-by-(locally finite). In particular, $IAut(A)$ has a normal subgroup $Γ$ such that $IAut(A)/Γ$ is locally finite and $Γ'$ is an abelian periodic subgroup whose all subgroups are normal in $Γ$. In the case when $A$ is periodic, $IAut(A)$ results to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of $IAut(A)$ in some other cases for $A$.