Automorphisms of a distance-regular graph with intersection array {176, 135, 32, 1; 1, 16, 135, 176}

A distance-regular graph $\Gamma$ with intersection array $\{176,135,32,1;1,16,135,176\}$ is an $AT4$-graph. Its antipodal quotient $\bar\Gamma$ is a strongly regular graph with parameters $(672,176$, $40,48)$. In both graphs the neighborhoods of vertices are strongly regular with parameters $(176,40,12,8)$. We study the automorphisms of these graphs. In particular, the graph $\Gamma$ is not arc-transitive. If $G=\mathrm{Aut}\,(\Gamma)$ contains an element of order 11, acts transitively on the vertex set of $\Gamma$, and $S(G)$ fixes each antipodal class, then the full preimage of the group $(G/S(G))'$ is an extension of a group of order 3 by $M_{22}$ or $U_6(2)$. We describe automorphism groups of strongly regular graphs with parameters $(176,40,12,8)$ and $(672,176,40,48)$ in the vertex-symmetric case.