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

We consider $G=Q_8,S{D}_{16},G_{24},$ 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 $(\ast -{\sigma }_i)$-graded $Q_8$- and $S{D}_{16}$-homotopy fixed point spectral sequences, where ${\sigma }_i$ is a non-trivial one-dimensional representation of $Q_8$.