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

We construct families of graphs from linear groups \(\mathrm{SL}(2,q)\), \(\mathrm{GL}(2,q)\) and \(\mathrm{GU}(2,q^2)\), 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.