On the maximum intersecting sets of the general semilinear group of degree $2$

Let $p$ be a prime and $q=p^k$. A subset $\mathcal{F}\subset {\mathrm{\Gamma L}}_2(q)$ is intersecting if any two semilinear transformations in $\mathcal{F}$ agree on some non-zero vector in ${\mathbb{F}}_q^2$. We show that any intersecting set of ${\mathrm{\Gamma L}}_2(q)$ is of size at most that of a stabilizer of a non-zero vector, and we characterize the intersecting sets of this size. Our proof relies on finding a subgraph which is a lexicographic product in the derangement graph of ${\mathrm{\Gamma L}}_2(q)$ in its action on non-zero vectors of ${\mathbb{F}}_q^2$. This method is also applied to give a new proof that the only maximal intersecting sets of ${\mathrm{GL}}_2(q)$ are the maximum intersecting sets.