The present paper is devoted classification of
A
T
4
(
p
,
p
+
2
,
r
)
\mathrm {AT4}(p,p+2,r)
-graphs. There is a unique
A
T
4
(
p
,
p
+
2
,
r
)
\mathrm {AT4}(p,p+2,r)
-graph with
p
=
2
p=2
, namely, the distance-transitive Soicher graph with intersection array
{
56
,
45
,
16
,
1
;
1
,
8
,
45
,
56
}
\{56, 45, 16, 1;1, 8, 45, 56\}
, whose local graphs are isomorphic to the Gewirtz graph. The existence of an
A
T
4
(
p
,
p
+
2
,
r
)
\mathrm {AT4}(p,p+2,r)
-graph with
p
>
2
{p>2}
remains an open question. It is known that the local graphs of each
A
T
4
(
p
,
p
+
2
,
r
)
\mathrm {AT4}(p,p+2,r)
-graph are strongly regular with parameters
(
(
p
+
2
)
(
p
2
+
4
p
+
2
)
,
p
(
p
+
3
)
,
p
−
2
,
p
)
\big ((p+2)(p^2+4p+2),p(p+3),p-2,p\big )
. In this paper, an upper bound is found for the prime spectrum of the automorphism group of a strongly regular graph with such parameters, and also some restrictions obtained for the prime spectrum and the structure of the automorphism group of an
A
T
4
(
p
,
p
+
2
,
r
)
\mathrm {AT4}(p,p+2,r)
-graph in the case when
2
>
p
2>p
is a prime power. As a corollary, it is shown that there are no arc-transitive
A
T
(
p
,
p
+
2
,
r
)
\mathrm {AT}(p,p+2,r)
-graphs with
p
∈
{
11
,
17
,
27
}
p\in \{11,17,27\}
.