  Let $G$ be a finite group, $\Omega(G)$ be its Burnside ring, and $\Delta(G)$
its augmentation ideal. Denote by $\Delta^n(G)$ and $Q_n(G)$ the $n$-th power
of $\Delta(G)$ and the $n$-th consecutive quotient group
$\Delta^n(G)/\Delta^{n+1}(G)$, respectively. This paper provides an explicit
$\mathbb{Z}$-basis for $\Delta^n(\mathcal{H})$ and determine the isomorphism
class of $Q_n(\mathcal{H})$ for each positive integer $n$, where
$\mathcal{H}=\langle g,h |\, g^{p^m}=h^p=1, h^{-1}gh=g^{p^{m-1}+1}\rangle$, $p$
is an odd prime.
