Cohomology of the Morava stabilizer group through the duality resolution at $n=p=2$

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