  Demonet, Iyama and Jasso introduced a new class of finite dimensional
algebras, $\tau$-tilting finite algebras. It was shown by Eisele, Janssens and
Raedschelders that tame blocks of group algebras of finite groups are always
$\tau$-tilting finite. Given the classical result that the representation type
(representation finite, tame or wild) of blocks is determined by their defect
groups, it is natural to ask what kinds of subgroups control $\tau$-tilting
finiteness of group algebras or their blocks. In this paper, as a positive
answer to this question, we demonstrate that $\tau$-tilting finiteness of a
group algebra of a finite group $G$ is controlled by a $p$-hyperfocal subgroup
of $G$ under some assumptions on $G$. We consider a group algebra of a finite
group $P\rtimes H$ over an algebraically closed field of positive
characteristic $p$, where $P$ is an abelian $p$-group and $H$ is an abelian
$p'$-group acting on $P$, and show that $p$-hyperfocal subgroups determine
$\tau$-tilting finiteness of the group algebras in this case.
