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In many categories, such as the category of graphs or topological spaces, every group appears as an automorphism group of an object in that category. This certainly isn't true for all categories, even in cases when one might expect it; see Is every group the automorphism group of a group? for an example. Is there any category (ideally one that people care about) where it is unknown whether this property holds, or at least, a category where the collection of groups which do appear as automorphism groups is poorly understood?

Beren Gunsolus
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    This is a wild guess: what is known about the groups that are the automorphism groups in the subcategory of the homotopy category comprising connected spaces? – Rob Arthan Mar 30 '20 at 23:27
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    @RobArthan I think every group is an automorphism group of a connected space in the homotopy category. The automorphism group of $BG$, the classifying space of a group $G$, is the outer automorphism group of $G$, and it is known that every group is an outer automorphism group. – Jeremy Rickard Mar 31 '20 at 08:20
  • I think that in the category of number fields this question is an active research topic, see 'inverse Galois theory'. – sss89 Mar 31 '20 at 19:12
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    For the full subcategory of the category of fields under $\mathbb Q$ (category of all $\mathbb Q \rightarrow F$) where $\mathbb Q \rightarrow F$ is galois, this is known as the inverse galois problem. – Noel Lundström Mar 31 '20 at 20:12
  • @JeremyRickard: thanks for the correction. I didn't know that every group is an outer automorphism group. Perhaps you could post a reference. – Rob Arthan Mar 31 '20 at 23:16
  • @RobArthan T. Matumoto, Any group is represented by an outer automorphism group, Hiroshima Math. J. 19 (1) (1989) 209–219. – Jeremy Rickard Apr 01 '20 at 06:07
  • @JeremyRickard: thanks! – Rob Arthan Apr 01 '20 at 20:27

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