When the automorphism group of an object determines the object

The following result holds.

Theorem.

(1) $\,$ (Baer-Kaplanski) $\,$ If $G$ and $H$ are torsion groups with isomorphic endomorphism rings $\mathrm{End}(G)$ and $\mathrm{End}(H)$, then $G$ and $H$ are isomorphic, and any ring isomorphism $\psi \colon \mathrm{End}(G) \to \mathrm{End}(H)$ is induced by some group isomorphism $\varphi \colon G \to H$.

(2) $\,$ (Leptin-Liebert) If $G$, $H$ are abelian $p$-groups $(p >3)$ and $\mathrm{Aut}(G)$ is isomorphic to $\mathrm{Aut}(H)$, then $G$ is isomorphic to $H$.

See A. V. Mikhalev, G. Pilz: The Concise Handbook of Algebra, p.74 and the references given therein.

If $M,N$ are two countable, $\omega$-categorical and $\omega$-stable structures, and $\operatorname{Aut}(M)\cong \operatorname{Aut}(N)$ (as topological groups), then $M$ and $N$ are bi-interpretable.

More generally, $M$ and $N$ only need to be countable, $\omega$-categorical and satisfy the so-called small index property. As far as I can recall, an analogous result is true for metric structures (in continuous logic).

Under the Generalized Continuum Hypothesis, $$2^{\aleph_\alpha}=\aleph_{\alpha+1}\quad(\forall\alpha),$$ sets with no structure (so automorphisms are just bijections) is an example.

Namely, by

Cardinality of the permutations of an infinite set

we have $\text{card Aut}(X)=2^{\text{card}(X)}$, and $$2^{\aleph_\alpha}=2^{\aleph_\beta}\implies\aleph_{\alpha+1}=\aleph_{\beta+1}\implies\alpha+1=\beta+1\implies\alpha=\beta.$$

(I suppose it's easy to check that we need more than ZFC but do not need GCH here.)