Is there a quadrangle $Q \subset \Bbb CP^2$, namely $Q$ is a set of four points, such that every permutation of $Q$ can be realizad by an isometric projectivity of $\Bbb CP^2$? Clearly the analogous question for a triangle in $\Bbb CP^1 \cong S^2$ has affirmative answer: it is a maximal equilateral triangle.

More generally, is there a subset of $n+2$ points in $\Bbb CP^n$ such that all of its permutations are restrictions of some isometric projectivity?