All extensions of $C_2$ by $C_{2^n} \times C_{2^n}$ are good for the Morava $K$-theory

Abstract

Let $C_m$ be a cyclic group of order $m$. We prove that if a group $G$ fits into an extension $1 \rightarrow C^2_{2^{n+1}} \rightarrow G \rightarrow C_2 \rightarrow 1$ for $n\geq 1$ then $G$ is good in the sense of Hopkins-Kuhn-Ravenel, i.e., $K(s)^\ast(BG)$ is evenly generated by transfers of Euler classes of complex representations of subgroups of $G$.