Automorphisms of a strongly regular graph with parameters $(532,156,30,52)$

Prime divisors of orders of automorphisms and the fixed point subgraphs of automorphisms of prime orders are studied for a hypothetical strongly regular graph with parameters $(532,156,30,52)$. Let $\Gamma$ be a strongly regular graph with parameters $(532,156,30,52)$ and $G={\rm Aut}(\Gamma)$ be a nonsolvable group acting transitively on the vertex set of $\Gamma$. Then $\bar G=G/O_2(G)\cong J_1$, $S(G)=O_2(G)$ is an irreducible $F_2J_1$-module, $|O_2(G)|>2$ and $\bar G_a\cong L_2(11)$.