  We consider $G=Q_8,SD_{16},G_{24},$ and $G_{48}$ as finite subgroups of the
Morava stabilizer group which acts on the height $2$ Morava $E$-theory
$\mathbf{E}_2$ at the prime $2$. We completely compute the $G$-homotopy fixed
point spectral sequences of $\mathbf{E}_2$. Our computation uses recently
developed equivariant techniques since Hill, Hopkins, and Ravenel. We also
compute the $(*-\sigma_i)$-graded $Q_8$- and $SD_{16}$-homotopy fixed point
spectral sequences, where $\sigma_i$ is a non-trivial one-dimensional
representation of $Q_8$.
