Let $M$ and $N$ be two first order structures, say they are countable and $\aleph_0$-categorical. Then $M$ and $N$ are bi-interpretable if and only if their automorphism groups $Aut(M)$ and $Aut(N)$ are isomorphic as topological groups.
Is the same true in general? What if $M$ and $N$ are uncountably categorical.