On the automorphism group of a putative Conway 99-graph

Let $\mathrm{\Gamma }$ be a {Conway 99-graph}, that is, a strongly regular graph with parameters $(99,14,1,2)$. In Makhnev and Minakova (On automorphisms of strongly regular graphs with parameters $\lambda =1$, $\mu =2$, Discrete Math.\ Appl.\ 14 (2) (2004) 201-210), the authors prove that the automorphism group $G$ of $\mathrm{\Gamma }$ must have order dividing $2\cdot 3^3\cdot 7\cdot 11$. They further show that if $\mathrm{\mid }G\mathrm{\mid }$ is divisible by $2$ then $\mathrm{\mid }G\mathrm{\mid }$ must divide $42$. In the present paper, we refine these results by proving that divisibility by $7$ implies $G\cong {\mathbb{Z}}_7$. As a consequence, divisibility by $2$ implies $\mathrm{\mid }G\mathrm{\mid }$ divides $6$, \ie $G$ is isomorphic to one of ${\mathbb{Z}}_2,{\mathbb{Z}}_6,S_3$.