It is usually said that groups can (or should) be thought of as "symmetries of things". The reason is that the "things" which we study in mathematics usually form a category and for every object $X$ of a (locally small) category $\mathcal{C}$, the set of automorphisms (symmetries) of $X$, denoted by $\text{Aut}_{\mathcal{C}}(X)$, forms a group.
My question is: Which categories that occur naturally in mathematics admit all kinds of symmetries? More precisely, for which categories we can solve the equation (of course up to isomorphism) $$\text{Aut}_{\mathcal{C}}(X) = G$$ for every group $G$?
I will write what I could find myself about this, which also hopefully illustrates what kind of answers that would interest me:
Negative for $\mathsf{Set}$: Infinite sets have infinite symmetry groups and for finite sets we get $S_n$'s. So if we let $G$ to be any finite group which is not isomorphic to some $S_n$, the equation has no solution. Negative for $\mathsf{Grp}$: No group can have its automorphism group a cyclic group of odd order. Positive for $\mathsf{Grph}$ (category of graphs): Frucht's theorem settles this for finite groups. Also according to the wikipedia page, the general situation was solved independently by de Groot and Sabidussi. An obvious necessary condition is that $\mathcal{C}$ should be a large category. This paper shows that the equation can be solved if $\mathcal{C}$ is the category of Riemann surfaces with holomorphic mappings and $G$ is countable. If we take $\mathcal{C}$ to be the category of fields with zero characteristic, I guess the equation relates to the inverse Galois problem. Edit: This may be much easier than the inverse Galois problem, as Martin Brandenburg commented.
group-theory
category-theory
big-list