Say that two structures $\mathfrak{A},\mathfrak{B}$ in possibly different finite languages are parametrically equivalent - and write "$\mathfrak{A}\approx\mathfrak{B}$" - iff they have the same underlying set and the same set of definable-with-parameters relations. I'm curious about the "automorphism group spectrum" of a parametric equivalence class. Specifically, for a structure $\mathfrak{A}$ let $$\mathsf{AGS}(\mathfrak{A})=\{\operatorname{Aut}(\mathfrak{B}):\mathfrak{B}\approx\mathfrak{A}\}.$$

This family of groups is a poset with respect to inclusion. Here the "same underlying set" condition is crucial: $\mathsf{AGS}(\mathfrak{A})$ is a family of subgroups of the full permutation group of that underlying set, and I am not identifying isomorphic elements of $\mathsf{AGS}(\mathfrak{A})$ or requiring the $\mathfrak{B}$s considered in forming $\mathsf{AGS}(\mathfrak{A})$ to be in the same signature as $\mathfrak{A}$ itself.

There is a lot of potential flexibility here. First of all, even if $\mathfrak{A}$ is pointwise-definable (so that $\operatorname{Aut}(\mathfrak{A})$ is trivial a fortiori) the set $\mathsf{AGS}(\mathfrak{A})$ may be quite rich: in particular, Matt F. showed that the structure $\mathfrak{Q}$ considered below is parametrically equivalent to a structure whose automorphism group acts $1$-transitively. A bit more surprisingly (to me at least), at MSE Harry West showed that $\mathsf{AGS}(\mathfrak{A})$ need not be upwards-directed.

But that's about all I know. In particular, I don't know much at all about computing $\mathsf{AGS}(\mathfrak{A})$ even for relatively simple interesting structures. I'd like to rectify this. To narrow the field (hehe), I'll focus on the following two candidates:

• The field of rationals $\mathfrak{Q}=(\mathbb{Q};+,\times)$.

• The field of reals $\mathfrak{R}=(\mathbb{R};+,\times)$.

Model-theoretically, $\mathfrak{Q}$ is quite wild (e.g. bi-interpretable with the integers) while $\mathfrak{R}$ is quite tame (e.g. decidable and o-minimal). It's not clear to me which of tameness or wildness makes $\mathsf{AGS}$ easier to analyze, but I suspect that at least one of these has some easily-establishable "coarse" properties (originally I just asked about $\mathfrak{Q}$). That said, all of this may be a red herring: as noted above, even pointwise-definability isn't particularly relevant to $\mathsf{AGS}$, so the model-theoretic dividing lines I'm used to might not be relevant at all.

Anyways:

Problem: describe at least one of $\mathsf{AGS}(\mathfrak{Q})$ or $\mathsf{AGS}(\mathfrak{R})$.

I'm especially interested in the question of whether each is upwards-directed.