A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut( $$Fi_{22}$$ ) Which Has a Nontrivial Stabilizer of a Ball of Radius $$2$$

Earlier, to confirm that one of the possibilities for the structure of vertex stabilizers of graphs with projective suborbits is realizable, we announced the existence of a connected graph \(\Gamma\) admitting a group of automorphisms \(G\) which is isomorphic to Aut\((Fi_{22})\) and has the following properties. First, the group \(G\) acts transitively on the set of vertices of \(\Gamma\), but intransitively on the set of \(3\)-arcs of \(\Gamma\). Second, the stabilizer in \(G\) of a vertex of \(\Gamma\) induces on the neighborhood of this vertex a group \(PSL_{3}(3)\) in its natural doubly transitive action. Third, the pointwise stabilizer in \(G\) of a ball of radius 2 in \(\Gamma\) is nontrivial. In this paper, we construct such a graph \(\Gamma\) with \(G=\mathrm{Aut}(\Gamma)\).