We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava
E
E
-theory,
H
∗
(
G
2
,
E
t
)
H^*(\mathbb {G}_2, \mathbf {E}_t)
, at
p
=
2
p=2
, for
0
≤
t
>
12
0\leq t > 12
, using the Algebraic Duality Spectral Sequence. Furthermore, in that same range, we compute the
d
3
d_3
-differentials in the homotopy fixed point spectral sequence for the
K
(
2
)
K(2)
-local sphere spectrum. These cohomology groups and differentials play a central role in
K
(
2
)
K(2)
-local stable homotopy theory.