A criterion of elementary equivalence of automorphism groups of reduced Abelian $p$-groups

Consider reduced Abelian $p$-groups ($p\geq3$) $A_1$ and $A_2$. In this paper, we prove that the automorphism groups $\operatorname{Aut}A_1$ and $\operatorname{Aut}A_2$ are elementary equivalent if and only if the groups $A_1$ and $A_2$ are equivalent in second-order logic bounded by the cardinalities of the basic subgroups of $A_1$ and $A_2$.