Automorphisms of the $AT4(6,6,3)$-graph and its strongly-regular graphs

Koolen and Jurisich defined class of $AT4$-graphs (tight antipodal graph of diameter $4$). Among these graphs available graph with intersection array $\{288,245,48,1;1,24,245,288\}$ on $v=1+288+2940+576+2=3807$ vertices. Antipodal quotient of this graph is strongly regular graph with parameters $(1269,288,42,72)$. Both these graphs are locally pseudo $GQ(7,5)$-graphs. In this paper we find possible automorphisms of these graphs. In particular, group of automorphisms of distance-regular graph with intersection array $\{288,245,48,1;1,24,245,288\}$ acts intransitive on the set of its antipodal classes.