A class of finite $p$-groups and the normalized unit groups of group algebras

Let $p$ be a prime and ${\mathbb{F}}_p$ be a finite field of $p$ elements. Let ${\mathbb{F}}_{p}G$ denote the group algebra of the finite $p$-group $G$ over the field ${\mathbb{F}}_p$ and $V({\mathbb{F}}_{p}G)$ denote the group of normalized units in ${\mathbb{F}}_{p}G$. Suppose that $G$ is a finite $p$-group given by a central extension of the form $$1⟶{\mathbb{Z}}_{p^n}\times {\mathbb{Z}}_{p^m}⟶G⟶{\mathbb{Z}}_p\times \cdots \times {\mathbb{Z}}_p⟶1$$ and $G^{\mathrm{\prime }}\cong {\mathbb{Z}}_p$, $n,m\ge 1$ and $p$ is odd. In this paper, the structure of $G$ is determined. And the relations of $V({\mathbb{F}}_{p}G{)}^{p^l}$ and $G^{p^l}$, ${\mathrm{\Omega }}_l(V({\mathbb{F}}_{p}G))$ and ${\mathrm{\Omega }}_l(G)$ are given. Furthermore, there is a direct proof for $V({\mathbb{F}}_{p}G{)}^p\bigcap G=G^p$.