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

We consider Abelian $p$-groups ($p\geq3$) $A_1=D_1\oplus G_1$ and $A_2=D_2\oplus G_2$, where $D_1$ and $D_2$ are divisible and $G_1$ and $G_2$ are reduced subgroups. We prove that if the automorphism groups $\operatorname{Aut}A_1$ and $\operatorname{Aut}A_2$ are elementarily equivalent, then the groups $D_1$, $D_2$ and $G_1$, $G_2$ are equivalent, respectively, in the second-order logic.