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 $g_1,g_2 \in S$, there exists $\alpha \in PG(1,q)$ such that $\alpha^{g_1}= \alpha^{g_2}$. 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 $S \subset PGL(2,q)$ 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.