The automorphism group of the derangement graph of $\operatorname{PGL}_{2}(q)$ acting on the projective line

Given a finite transitive group $G\leq \operatorname{Sym}(Ω)$, the derangement graph $Γ_G$ is the graph whose vertex set is $G$, and two vertices $g$ and $h$ are adjacent if the ratio $h^{-1}g$ is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement g