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Let $p$ be a prime and $F$ be a finite field of characteristic $p$. Supposethat $FG$ 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. Inthis 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, theorder of $V_*(FG)$ can be divisible by$|F|^{\frac{1}{2}(|G|+|\Omega_1(G)|)-1}$, where $\Omega_1(G)=\{g\in G\ |\g^2=1\}$.

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