The Conjecture: If $H$ is characteristic in $G$ and $K\leq G$ such that $K\cong H$, then $H=K$.
We know that $\sigma(H)=H$ for all $\sigma\in\text{Aut}(G)$. Hence the a more general question here is: does every isomorphism $\tau: H\to K$ between two subgroups $H$ and $K$ extend to an automorphism of $G$? If yes, then the conjecture will hold immediately.
Any confirmation would be greatly appreciated.