Elementary equivalence of automorphism groups of reduced Abelian $p$-groups

Unbounded reduced Abelian $p$-groups ($p\geq3$) $A_1$ and $A_2$ are considered. It is proved that if the automorphism groups $\operatorname{Aut}A_1$ and $\operatorname{Aut}A_2$ are elementary equivalent, then the groups $A_1$ and $A_2$ are equivalent in the second order logic bounded with the final rank of the basic subgroups of $A_1$ and $A_2$.