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.If A is any periodic abelian group, then PAut(A) is the cartesian product of all theAccording to [10], an automorphism γ is called an (invertible) multiplication of A if and only if it is a power automorphism of A, if A is periodic, or -when A is non-periodicthere exist coprime integers m, n such that (na)γ = ma, for each a ∈ A. In the latter case, we have mnA = A and A π(mn) = 0 and -with an abuse of notation-we will write γ = m/n. We warn that we are using the word "multiplication" in a way different from [14]. Invertible multiplications of A form a subgroup which is a central subgroup of Aut(A).