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:

> **Question**: What exactly is $\mathsf{AGS}(\mathbb{Q};+,\times)$? In particular, is it upwards-directed?