Categories where possible automorphism groups are not understood.

Question

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?