  The \textit{transition matrix} of a graph $\Gamma$ with adjacency matrix $A$
is defined by $H(\tau ) := \exp(-\mathbf{i}\tau A)$, where $\tau \in
\mathbb{R}$ and $\mathbf{i} = \sqrt{-1}$. The graph $\Gamma$ exhibits
\textit{perfect state transfer} (PST) between the vertices $u$ and $v$ if there
exists $\tau_0(>0)\in \mathbb{R}$ such that $\lvert H(\tau_0)_{uv} \rvert = 1$.
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 $\text{Cay}(V_{8n},
S)$.
