Summary: We consider the action of the \(2\)-dimensional projective general linear group \(\mathrm{PGL}(2,q)\) on the projective line \(\mathrm{PG}(1,q)\). A subset \(S\) of \(\mathrm{PGL}(2,q)\) is said to be an intersecting family if for every \(g_1,g_2 \in S\), there exists \(\alpha \in\mathrm{PG}(1,q)\) such that \(\alpha^{g_1}= \alpha^{g_2}\). It was proved by K. Meagher and P. Spiga [J. Comb. Theory, Ser. A 118, No. 2, 532–544 (2011; Zbl 1227.05163)] that the intersecting families of maximum size in \(\mathrm{PGL}(2,q)\) are precisely the cosets of point stabilizers. We prove that if an intersecting family \(S \subset \mathrm{PGL}(2,q)\) 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.