On distance-regular graphs with intersection arrays $\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}$

If a distance-regular graph $\Gamma$ of diameter 3 contains a maximal locally regular 1-code that is last subconstituent perfect, then $\Gamma$ has intersection array $\{a(p+1),cp,a+1;1,c,ap\}$ or $\{a(p+1),(a+1)p,c;1,c,ap\}$, where $a=a_3$, $c=c_2$, and $p=p^3_{33}$ (Jurišić, Vidali). In the first case, $\Gamma$ has eigenvalue $\theta_2=-1$ and the graph $\Gamma_3$ is pseudogeometric for $GQ(p+1,a)$. If $a=c+1$, then the graph $\bar\Gamma_2$ is pseudogeometric for $pG_2(p+1,2a)$. If in this case the pseudogeometric graph for the generalized quadrangle $GQ(p+1,a)$ has quasi-classical parameters, then $\Gamma$ has intersection array $\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}$ (Makhnev, Nirova). In this paper, we find possible automorphisms of a graph with intersection array $\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}$.