Let
m
m
be an integer greater than three and
ℓ
\ell
be an odd prime. In this paper we prove that at least one of the following groups:
P
Ω
2
m
±
(
F
ℓ
s
)
\mathrm {P}\Omega ^\pm _{2m}(\mathbb {F}_{\ell ^s})
,
P
S
O
2
m
±
(
F
ℓ
s
)
\mathrm {PSO}^\pm _{2m}(\mathbb {F}_{\ell ^s})
,
P
O
2
m
±
(
F
ℓ
s
)
\mathrm {PO}_{2m}^\pm (\mathbb {F}_{\ell ^s})
, or
P
G
O
2
m
±
(
F
ℓ
s
)
\mathrm {PGO}^\pm _{2m}(\mathbb {F}_{\ell ^s})
is a Galois group of
Q
\mathbb {Q}
for infinitely many integers
s
>
0
s > 0
. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen, and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of
G
L
2
m
(
A
Q
)
\mathrm {GL}_{2m}(\mathbb {A}_\mathbb {Q})
.