  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 $\Omega_1(G)=\{g\in G\ |\
g^2=1\}$.
