  In this paper, we show that weakly symmetric $\tau$-tilting finite algebras
have positive definite Cartan matrices, which implies that we can prove
$\tau$-tilting infiniteness of weakly symmetric algebras by calculating their
Cartan matrices. Similarly, we obtain the condition on Cartan matrices that
selfinjective algebras are $\tau$-tilting infinite. By applying this result, we
show that a group algebra of $(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is
$\tau$-tilting infinite when $p^l\geq n$ and
$\#\mathrm{IBr}\,H\geq\min\{p,3\}$, where $p>0$ is the characteristic of the
ground field, $H$ is a subgroup of the symmetric group $\mathfrak{S}_n$ of
degree $n$, the action of $H$ permutes the entries of
$(\mathbb{Z}/p^l\mathbb{Z})^n$, and $\mathrm{IBr}\,H$ denotes the set of
irreducible $p$-Brauer characters of $H$. Moreover, we show that under the
assumption that $p^l\geq n$ and $H$ is a $p'$-subgroup of $\mathfrak{S}_n$,
$\tau$-tilting finiteness of a group algebra of a group
$(\mathbb{Z}/p^l\mathbb{Z})^n\rtimes H$ is determined by its $p$-hyperfocal
subgroup.
