For any abelian group $G$ ad any $n\in\mathbb N$, let $G[n]$ denote the subgroup $\{g\in G\mid ng=0\}$. Now suppose $G$ is a $p$-primary group for some prime number $p$ and $H$ $H^\prime$ two pure subgroups of $G$. My question is:
If $H[p]=H^\prime[p]$, can we deduce that there is an automorphism $f:G\overset{\sim}{\to} G$ which maps $H$ onto $H^\prime$ ?
I can prove that this is the case when either of $H$ and $H^\prime$ is a direct summand, and that if $H\subseteq H^\prime$, then $H=H^\prime$.