Altough this sounds as a very basic question, I didn't receive any answer on stack exchange and by people more knowledgeable than me
Take $p$ a prime number and $P$ an abelian finite $p$-group. Let $A,A'$ be subgroup of $P$ such that $A \simeq A'$ and $P/A \simeq P/A'$ as groups. Can I conclude that there is $\phi \in Aut(P)$ such that $\phi(A)=A'$?
Thanks in advance!
p.s:https://math.stackexchange.com/questions/1476641/obstruction-to-be-conjugated-by-an-automorphism-for-subgroups-of-an-abelian-grou, here the MSE link