  This paper is devoted to the problem of classification of ${\rm
AT4}(p,p+2,r)$-graphs. There is a unique ${\rm AT4}(p,p+2,r)$-graph with $p=2$,
namely, the distance-transitive Soicher graph with intersection array $\{56,
45, 16, 1;1, 8, 45, 56\}$, whose local graphs are isomorphic to the Gewirtz
graph. It is still unknown whether an ${\rm AT4}(p,p+2,r)$-graph with $p>2$
exists. The local graphs of each ${\rm AT4}(p,p+2,r)$-graph are strongly
regular with parameters $((p+2)(p^2+4p+2),p(p+3),p-2,p)$. In the present paper,
we find an upper bound for the prime spectrum of an automorphism group of a
strongly regular graph with such parameters, and we also obtain some
restrictions for the prime spectrum and the structure of an automorphism group
of an ${\rm AT4}(p,p+2,r)$-graph in case when $p$ is a prime power. As a
corollary, we show that there are no arc-transitive ${\rm AT4}(p,p+2,r)$-graphs
with $p\in \{11,17,27\}$.
