  Suppose $G$ is a finitely generated group and $H$ is a subgroup of $G$. Let
$\partial_{c}^{\mathcal{F}\mathcal{Q}}G$ denote the contracting boundary of $G$
with the topology of fellow travelling quasi-geodesics defined by Cashen-Mackay
\cite{cashen2017}.
  In this article, we show that if the limit set $\Lambda(H)$ of $H$ in
$\partial_{c}^{\mathcal{F}\mathcal{Q}}G$ is compact and contains at least three
points then the action of the subgroup $H$ on the space of distinct triples
$\Theta_{3}(\Lambda(H))$ is properly discontinuous. By applying a result of B.
Sun \cite{BinSun}, if the limit set $\Lambda(H)$ is compact and the action of
$H$ on $\partial_{c}^{\mathcal{F}\mathcal{Q}}G$ is non-elementary then $H$
becomes an acylindrically hyperbolic group
