Morava $K$-theory rings of the extensions of $C_2$ by the products of cyclic $2$-groups

In 2011, Schuster proved that $\mod2$ Morava $K$-theory $K(s)^*(BG)$ is evenly generated for all groups $G$ of order $32$. There exist $51$ non-isomorphic groups of order $32$. In a monograph by Hall and Senior, these groups are numbered by $1,\dots,51$. For the groups $G_{38},\dots,G_{41}$, which fit in the title, the explicit ring structure is determined in a joint work of M. Jibladze and the author. In particular, $K(s)^*(BG)$ is the quotient of a polynomial ring in 6 variables over $K(s)^*(\mathrm{pt})$ by an ideal generated by explicit polynomials. In this article we present some calculations using the same arguments in combination with a theorem by the author on good groups in the sense of Hopkins–Kuhn–Ravenel. In particular, we consider the groups $G_{36},G_{37}$, each isomorphic to a semidirect product $(C_4\times C_2\times C_2)\rtimes C_2$, the group $G_{34}\cong(C_4\times C_4)\rtimes C_2$ and its non-split version $G_{35}$. For these groups the action of $C_2$ is diagonal, i.e., simpler than for the groups $G_{38},\dots,G_{41}$, however the rings $K(s)^*(BG)$ have the same complexity.