RO(G)-Graded Homotopy Fixed Point Spectral Sequence for Height 2 Morava E-Theory

We consider $G=Q_8,{\text{SD}}_{16},G_{24},$ and $G_{48}$ as finite subgroups of the Morava stabilizer group which acts on the height 2 Morava E-theory $E_2$ at the prime 2. We completely compute the G-homotopy fixed point spectral sequences of $E_2$. Our computation uses recently developed equivariant techniques since Hill, Hopkins, and Ravenel. We also compute the $(\ast -{\sigma }_i)$-graded $Q_8$- and ${\text{SD}}_{16}$-homotopy fixed point spectral sequences, where ${\sigma }_i$ is a non-trivial one-dimensional representation of $Q_8$.