Let
Γ
0
(
n
,
N
)
\Gamma _0(n,N)
denote the usual congruence subgroup of type
Γ
0
\Gamma _0
and level
N
N
in
SL
(
n
,
Z
)
\text {SL}(n,{\mathbb Z})
. Suppose for
i
=
1
,
2
i=1,2
that we have an irreducible odd
n
n
-dimensional Galois representation
ρ
i
\rho _i
attached to a homology Hecke eigenclass in
H
∗
(
Γ
0
(
n
,
N
i
)
,
M
i
)
H_*(\Gamma _0(n,N_i),M_i)
, where the level
N
i
N_i
and the weight and nebentype making up
M
i
M_i
are as predicted by the Serre-style conjecture of Ash, Doud, Pollack, and Sinnott. We assume that
n
n
is odd, that
N
1
N
2
N_1N_2
is squarefree, and that
ρ
1
⊕
ρ
2
\rho _1\oplus \rho _2
is odd. We prove two theorems that assert that
ρ
1
⊕
ρ
2
\rho _1\oplus \rho _2
is attached to a homology Hecke eigenclass in
H
∗
(
Γ
0
(
2
n
,
N
)
,
M
)
H_*(\Gamma _0(2n,N),M)
, where
N
N
and
M
M
are as predicted by the Serre-style conjecture. The first theorem requires the hypothesis that the highest weights of
M
1
M_1
and
M
2
M_2
are small in a certain sense. The second theorem requires the truth of a conjecture as to what degrees of homology can support Hecke eigenclasses with irreducible Galois representations attached, but no hypothesis on the highest weights of
M
1
M_1
and
M
2
M_2
. This conjecture is known to be true for
n
=
3
n=3
, so we obtain unconditional results for
GL
(
6
)
\text {GL}(6)
. A similar result for
GL
(
4
)
\text {GL}(4)
appeared in an earlier paper.