Stability for intersecting families in PGL(2, q)

We consider the action of the 2 -dimensional projective general linear group PGL(2,q) on the projective line PG(1, q). A subset S of PGL(2,q) is said to be an intersecting family if for every g(1), g(2) is an element of S, there exists alpha is an element of PG(1, q) such that alpha(g1) = alpha(g2). It was proved by Meagher and Spiga that the intersecting families of maximum size in PGL(2, q) are precisely the cosets of point stabilizers. We prove that if an intersecting family S subset of PGL(2, q) has size close to the maximum then it must be "close" in structure to a coset of a point stabilizer. This phenomenon is known as stability. We use this stability result proved here to show that if the size of S is close enough to the maximum then S must be contained in a coset of a point stabilizer.