  We show that for every prime $r$ all $r$-subgroups in the normalized units of
the integral group ring of $\operatorname{PSL}(2,p^3)$ are isomorphic to
subgroups of $\operatorname{PSL}(2,p^3)$. This answers a question of M.
Hertweck, C.R. H\"ofert and W. Kimmerle for this series of groups.
