Let
Γ
\Gamma
be a finite directed graph with vertex set
V
(
Γ
)
V(\Gamma )
and edge set
E
(
Γ
)
E(\Gamma )
and let G be a subgroup of
aut
(
Γ
)
{\operatorname {aut}}(\Gamma )
which we assume to act transitively on both
V
(
Γ
)
V(\Gamma )
and
E
(
Γ
)
E(\Gamma )
. Suppose that for some prime power q, the stabilizer
G
(
x
)
G(x)
of a vertex x induces on both
{
y
|
(
x
,
y
)
∈
E
(
Γ
)
}
\{ y|(x,y) \in E(\Gamma )\}
and
{
w
|
(
w
,
x
)
∈
E
(
Γ
)
}
\{ w|(w,x) \in E(\Gamma )\}
a group lying between
P
S
U
(
3
,
q
2
)
PSU(3,{q^2})
and
P
Γ
U
(
3
,
q
2
)
P\Gamma U(3,{q^2})
. It is shown that if G acts primitively on
V
(
Γ
)
V(\Gamma )
, then for each edge (x, y), the subgroup of
G
(
x
)
G(x)
fixing every vertex in
{
w
|
(
x
,
w
)
\{ w|(x,w)
or
(
y
,
w
)
∈
E
(
Γ
)
}
(y,w) \in E(\Gamma )\}
is trivial.