Given an action of a group \(G\) on a relational Fraïssé structure \(M\), we call this action sharply \(k\)-homogeneous if, for each isomorphism \( f : A \rightarrow B \) of substructures of \(M\) of size \(k\), there is exactly one element of \(G\) whose action extends \(f\). This generalises the well-known notion of a sharply \(k\)-transitive action on a set, and was previously investigated by Cameron, Macpherson and Cherlin.
I will discuss recent results with J. de la Nuez González which show that a wide variety of Fraïssé structures admit sharply \(k\)-homogeneous actions for \(k \leq 3\) by finitely generated virtually free groups. Our results also specialise to the case of sets, giving the first examples of finitely presented non-split infinite groups with sharply 2-transitive/sharply 3-transitive actions.