The intersection density of cubic arc-transitive graphs with $2$-arc-regular full automorphism group equal to $\operatorname{PGL}_2(q)$

The \emph{intersection density} of a transitive permutation group $G\leq \operatorname{Sym}(\Omega)$ is the ratio between the largest size of a subset of $G$ in which any two agree on at least one element of $\Omega$, and the order of a point-stabilizer of $G$. In this paper, we determine the intersection densities of the automorphism group of the arc-transitive graphs admitting a $2$-arc-regular full automorphism group $G^* = \operatorname{PGL}_2(q)$ and an arc-regular subgroup of automorphism $G = \operatorname{PSL}_2(q)$.