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

$RO(G)$-graded homotopy fixed point spectral sequence for height $2$ Morava $E$-theory