The unitary subgroups of group algebras of a class of finite $2$-groups with derived subgroup of order $2$

Abstract: Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Suppose that $FG$ is the group algebra of the finite $p$-group $G$ over the field $F$. Let $V(FG)$ denote the group of normalized units in $FG$ and let $V_(FG)$ denote the unitary subgroup of $V(FG)$. If $p$ is odd, then the order of $V_(FG)$ is $|F|^{(|G|-1)/2}$. However, the case when $p=2$ still is open. In this paper, the order of $V_(FG)$ is computed when $G$ is a nonabelian $2$-group given by a central extension of the form $$1\longrightarrow \mathbb{Z}{2^n}\times \mathbb{Z}{2^m} \longrightarrow G \longrightarrow \mathbb{Z}2\times \cdots\times \mathbb{Z}_2 \longrightarrow 1$$ and $G'\cong \mathbb{Z}_2$, $n, m\geq 1$. Further, a conjecture is confirmed, namely, the order of $V(FG)$ can be divisible by $|F|^{\frac{1}{2}(|G|+|\Omega_1(G)|)-1}$, where $\Omega_1(G)={g\in G\ |\ g^2=1}$.