Let $\Gamma$ be a strongly regular graph with parameters $(v,k,1,2)$. Then $k=u^2+u+2$ and $u=1,3,4,10$, or $31$. It is known that such graphs exist for $u$ equal to $1$ and $4$. They are the $(3\times 3)$-lattice and the graph of cosets of the ternary Golay code. If $u=3$, then $\Gamma$ has the parameters $(99,14,1,2)$. The question on existence of such graphs was posed by J. Seidel. With the use of theory of characters of finite groups we find the possible orders and the structures of subgraphs of the fixed points of automorphisms of the graph $\Gamma$ with parameters $(99,14,1,2)$. It is proved that if the group $\operatorname{Aut}(\Gamma)$ contains an involution, then its order divides $42$. This research was supported by the Russian Foundation for Basic Research, grant 02–01–00722.