I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that this understanding is the main goal (or the very definition) of Number Theory (give or take). Anyway, in this question I am concerned with *actions* of this group. I know that there is, for instance, Grothendieck's "Dessins d'enfants" theory (although Deligne has stated that, at the point of writing and publishing his famous "projective line minus three points paper," the dessins had not helped that understanding very much at all). If one would want to define some action of $L$ on some space/set/geometry/etc., what would make that action *interesting* ?

**Example given**: I have never seen descriptions of $L$ on combinatorial geometries such as (axiomatic) projective planes, but (when) would such actions be interesting enough ? What are natural constraints on the geometry acted upon, so that we would have a nice action ?