We study geometric monodromy groups
G
g
e
o
m
,
F
q
G_{{\mathrm {geom}},\mathcal {F}_q}
of the local systems
F
q
\mathcal {F}_q
on the affine line over
F
2
\mathbb {F}_2
of rank
D
=
q
/
2
(
q
−
1
)
D=\sqrt {q/2}(q-1)
,
q
=
2
2
n
+
1
q=2^{2n+1}
, constructed in N. Katz [Exponential sums, Ree groups and Suzuki groups: conjectures, Exp. Math. 28 (2019), 49-56.]. The main result of the paper shows that
G
g
e
o
m
,
F
q
G_{{\mathrm {geom}},\mathcal {F}_q}
is either the Suzuki simple group
2
B
2
(
q
)
{}^2 \! B_2(q)
, or the special linear group
S
L
D
\mathrm {SL}_D
. We also show that
F
8
\mathcal {F}_8
has geometric monodromy group
2
B
2
(
8
)
{}^2 \!B_2(8)
, and arithmetic monodromy group
A
u
t
(
2
B
2
(
8
)
)
\mathrm {Aut}({}^2 \! B_2(8))
over
F
2
\mathbb {F}_2
, thus establishing Katz’s Conjecture 2.2 in the above cited paper in the case
q
=
8
q=8
.