For the Lubotzky–Phillips–Sarnak graph $X^{p,q}$, what is its automorphism group? These graphs are not just ($p+1$)-regular but are Cayley graphs for $G=PSL_2(\mathbb{F}_q)$, so clearly $G\le Aut(G)$. But is there anything else known about the structure of the full automorphism group in this case? (As far as I can tell, the necessary condition of this post appears to be true here but the papers cited giving sufficiency do not apply – but I could be mistaken on this point.)

For the Lubotzky–Phillips–Sarnak (LPS) graph $X^{p,q}$, what is its automorphism group? These graphs are not just ($p+1$)-regular but are Cayley graphs for $G=\mathrm{PSL}_2(\mathbb{F}_q)$, so clearly $G\le \mathrm{Aut}(G)$. But is there anything else known about the structure of the full automorphism group in this case? (As far as I can tell, the necessary condition of this post appears to be true here but the papers cited giving sufficiency do not apply – but I could be mistaken on this point.)