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

Let p be a prime and q = pᵏ. A subset F ⊂ Γ L₂(q) is intersecting if any two semilinear transformations in F agree on some non-zero vector in Fq². We show that any intersecting set of Γ L₂(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 Γ L₂(q) in its action on non-zero vectors of Fq². This method is also applied to give a new proof that the only maximal intersecting sets of GL₂(q) are the maximum intersecting sets.