An Erdős–Ko–Rado theorem for the derangement graph of PGL(2,q) acting on the projective line

Let G=PGL(2,q) be the projective general linear group acting on the projective line Pq. A subset S of G is intersecting if for any pair of permutations π,σ in S, there is a projective point p∈Pq such that pπ=pσ. We prove that if S is intersecting, then |S|⩽q(q−1). Also, we prove that the only sets S that meet this bound are the cosets of the stabilizer of a point of Pq.