The \textit{transition matrix} of a graph $Γ$ with adjacency matrix $A$ is defined by $H(τ) := \exp(-\mathbf{i}τA)$, where $τ\in \mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$. The graph $Γ$ exhibits \textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there exists $τ_0(>0)\in \mathbb{R}$ such that $\lvert H(τ_0)_{uv} \rvert = 1$. For a positive integer $n$, the group $V_{8n}$ is defined as $V_{8n} := \langle a,b \colon a^{2n} = b^{4} = 1, ba = a^{-1}b^{-1}, b^{-1}a = a^{-1}b \rangle$. In this paper, we study the existence of perfect state transfer on Cayley graphs $\text{Cay}(V_{8n}, S)$. We present some necessary and sufficient conditions for the existence of perfect state transfer on $\text{Cay}(V_{8n}, S)$.