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 $G ={\rm Aut}(\Gamma)$.