which is isomorphic to $\mathbb{Z}_p^{\times}$, and $KU_p$ is the $p$ completion of periodic complex $K$-theory [@MR2030586]. Note that the Morava stabilizer group $\mathbb{G}_1=\mathbb{Z}_p^{\times}$ acts on $KU_p$ by Adams operations, so we may write $\psi_{q}$ for the Adams operation corresponding to the topological generator of $\mathbb{Z}_p^{\times}$ [@MR3328537]. The homotopy fixed points can therefore be modeled by the fiber sequence $$\xymatrix{ L_{K(1)}S \ar[r] & KU_p \ar[r]^{1- \psi_{q} } & KU_p. }$$ Under the stated conditions on $q$ and $p$, there is therefore a map of fiber sequences $$\xymatrix{
K(\mathbb{F}_q)_p \ar[r] \ar[d] & ku_p \ar[r]^{1-\psi_q} \ar[d] & \Sigma^2 ku_p \ar[d]\\
L_{K(1)}S \ar[r] & KU_p \ar[r]^{1-\psi_q} &  KU_p . }$$ By examining homotopy groups of each of the fibers, we see that the map of fibers induces an isomorphism in homotopy in degrees greater or equal to zero. Recall that the homotopy groups of $L_{K(1)}S$ are given by $$\pi_*(L_{K(1)}S) \cong \left \{ \begin{array}{ll} \mathbb{Z}_p & \text{ if } *=0,-1 \\ \mathbb{Z}_p/(q^i-1)\mathbb{Z}_p & \text{ if } *=2i-1 \text{ for } i \in \mathbb{Z} \end{array}\right.$$ and this agrees with Quillen’s computation $$K_*(\mathbb{F}_q) \cong \left \{ \begin{array}{ll} \mathbb{Z} & \text{ if } *=0  \\ \mathbb{Z}/(q^i-1)\mathbb{Z} & \text{ if } *=2i-1  \text{ for  } i \in \mathbb{N} \end{array} 