I am neither number theorist nor algebraic geometer. I am wondering whether Galois groups of number fields (say the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which are not related a priori to the number theory.

I am aware of two such situations of rather different nature:

(1) Grothendieck's dessins d'enfants: $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on certain graphs on 2-dimensional surfaces.

(2) $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the profinite completion of the topological $K$-theory (of sufficiently nice spaces, e.g. finite $CW$-complexes).

As far as I understand (am I wrong?) the most important and best studied examples of actions of Galois groups are actions on $l$-adic cohomology of varieties over number fields. But this is not what I am looking for: number fields appear in the formulation of the problem from the vary beginning.