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

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