Automorphism Groups of the $PS{L}_2(q)$ Commuting Involution Graphs

Given a finite group $G$ and a conjugacy class of involutions $X$ of $G$, we define the commuting involution graph $\mathcal{C}(G,X)$ to be the graph with vertex set $X$ and $x,y\in X$ adjacent if and only if $x\mathrm{\ne }y$ and $xy=yx$. In this paper the automorphism group of the graph $\mathcal{C}(G,X)$ is determined when $G=PS{L}_2(q)$.