  Manevitz and Weinberger (1996) proved that the existence of effective
$K$-Lipschitz $\mathbb{Z}/n\mathbb{Z}$-actions implies the existence of
effective $K$-Lipschitz $\mathbb{Q}/\mathbb{Z}$-actions for all compact
connected manifolds with metrics, where $K$ is a fixed Lipschitz constant. The
$\mathbb{Q}/\mathbb{Z}$-actions were constructed from suitable actions of a
sufficiently large hyperfinite cyclic group
${}^{\ast}{\mathbb{Z}}/\gamma{}^{\ast}{\mathbb{Z}}$ in the sense of nonstandard
analysis. By modifying their construction, we prove that for every direct
system $\left(\Lambda,G_{\lambda},i_{\lambda\mu}\right)$ of torsion groups with
monomorphisms, the existence of effective $K$-Lipschitz $G_{\lambda}$-actions
implies the existence of effective $K$-Lipschitz $\varinjlim
G_{\lambda}$-actions. This generalises Manevitz and Weinberger's result.
