Perfect state transfer in graphs related to linear groups in two dimensions

We construct families of graphs from linear groups $\mathrm{S}\mathrm{L}(2,q)$, $\mathrm{G}\mathrm{L}(2,q)$ and $\mathrm{G}\mathrm{U}(2,q^2)$, where $q$ is an odd prime power, with the property that the continuous-time quantum walks on the associated networks of qubits admit perfect state transfer.