TITLE: When the automorphism group of an object determines the object
QUESTION [74 upvotes]: Let me start with three examples to illustrate my question (probably vague; I apologize in advance).

$\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M, N\in \mathbf{Man}$ there is the following theorem (Whittaker) which says that 


$$\mathrm{Homeo}(M)\cong_{\text{as groups}} \mathrm{Homeo}(N) \, \textit{ if and only if } \, M\cong_{\text{as manifold}} N$$


$\mathbf{NFields}$, the category of Number fields. For any $F, K\in \mathbf{NFields}$  there is a theorem (Neukirch-Uchida) which says that 


$$\mathrm{Gal}(\overline{K}/K)\cong_{\text{as progroups}} \mathrm{Gal}(\overline{F}/F) \textit{ if and only if } K\cong_{\text{as fields}} F $$


$\mathbf{Vect}$, the category of finite dimensional vector spaces. For any $V, W\in \mathbf{Vect}$ we have that 


$$\mathrm{GL}(V)\cong \mathrm{GL}(W) \textit{ if and only if } V\cong W $$

In the first and the third examples we see that the automorphism group of an object determines the object. The second example seems to be similar in some sense however it does not admit the same naive interpretation.

$\textbf{Question:}$ 
  What are other non-trivial examples of interesting categories where the automorphism group of an object determines the object itself? Is there a name for such categories? Is there a way to compare and characterize these kind of categories?

REPLY [3 votes]: Geometric Complexity Theory:

Theorem (Mulmuley and Sohoni [MS])
  The permanent (respectively the determinant) polynomial is characterized by its symmetry group. 

That is if $P$ is a homogeneous polynomial of degree $m$ in $m^2$ variables and its symmetry group $G_P$ also fixes the permanent (respectively the determinant), then $P$ must be a scalar multiple of the permanent (respectively the determinant). 
Landsberg and Ressayre [LR] made progress on Valiant's version of P vs NP using this result.