Say that two structures $\mathfrak{A},\mathfrak{B}$ in finite languages $\Sigma,\Pi$ 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 **possible automorphism groups** of a parametric equivalence class. Specifically, for a structure $\mathfrak{A}$ let $$\mathsf{AGS}(\mathfrak{A})=\{\operatorname{Aut}(\mathfrak{B}):\mathfrak{B}\approx\mathfrak{A}\}.$$ (It's here that 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 $\mathsf{AGS}(\mathfrak{A})$.)


For example, [many rigid structures are](https://mathoverflow.net/questions/401321/is-mathbbq-equivalent-to-a-structure-with-transitive-automorphism-group-a) parametrically equivalent to structures with $1$-transitively-acting automorphism group. A bit more surprisingly (to me at least), at MSE user [Harry West showed](https://math.stackexchange.com/a/4220357/28111) that $\mathsf{AGS}(\mathfrak{A})$ need **not** be upwards-directed; however, the example he constructed was somewhat artificial. 

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 $\mathfrak{A}$. 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** $\mathfrak{R}=(\mathbb{R};+,\times)$.

Of course model-theoretically,  $\mathcal{Q}$ is quite wild (e.g. [bi-interpretable with the integers](http://www.math.umd.edu/~laskow/713/Spring17/carolslides.pdf)) while $\mathcal{R}$ 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 $\mathfrak{Q}$).

> **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.