This question is a follow-up to https://mathoverflow.net/questions/372349/are-there-infinitely-many-l-rigs?r=SearchResults and to https://mathoverflow.net/questions/355240/is-an-automorphic-form-of-operatornamegl-n-mathbba-mathbbq-deter.

I copy paste a deepl translation of an old answer I got to a question of mine on a French math forum:

"The idea is that to any "space" $X$ one can associate the category of its coverings $R(X)$. If we choose a "geometric point", we have a "fiber in $x$" functor: $\omega_{x}:R(X)\to Set$ which to a covering $f:Y\to X$ associates $f^{-1}(x)$. If we are given a path $\gamma:x_{0}\to x_{1}$, then, for any $y_{0}\in f^{-1}(x_{0})$, it rises to a single path $\tilde{\gamma}$ of origin $y_{0}$. Considering the end of this path, we get a point $y_{1}\in f^{-1}(x_{1})$. And this depends only on the homotopy class of $\gamma$. So we have an application that to a homotopy class of paths associates an isomorphism of fiber functors: $\pi_{1}(X;x_{0},x_{1})\to Isom(\omega_{x_{0}},\omega_{x_{1}})$. Grothendieck's brilliant remark is that this application is an isomorphism! In particular, $\pi_{1}(X,x)=Aut(\omega_{x})$ and the right-hand side term thus gives a purely algebraic definition (we do not use the topology of $\mathbb{R}$) of the fundamental group.

Where we join Galois theory is that, if we take - space $X$" = a field $K$ (we denote $X=Spec(K)$- "covering $Y$ of $X$" = finite separable extension $L/K$. - "geometric point" = embedding $x:K\to\Xi$ into an algebraically closed field. Then the associated "fiber in $x$" functor is the functor $ω_{x}:L/K\mapsto Hom_{K}(L,\Xi)$ which to $L/K$ associates its $K$-embeddings in $\Xi$. The theory applies and we find that the "fundamental group" of $K$ (i.e., the group of automorphisms of the fiber functor $\omega_{\Xi}$) is  the absolute Galois group $Gal(K^{sep}/K)$ of $K$ (Galois group of a separable closure)."

So I would like to know if, provided that an L-rig can be extended into a field, one can establish a parallel between the map that sends an automorphic representation of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ to the associated L-function and such a "fiber functor" that would allow to say that the automorphism group/"fundamental group" of $\mathcal{M}$ is an absolute Galois group. If yes, is it provably the absolute Galois group of the rationals?