Beauville surfaces and finite groups

Extending results of Bauer, Catanese and Grunewald, and of Fuertes and González-Diez, we show that Beauville surfaces of unmixed type can be obtained from the groups L2(q) and SL2(q) for all prime powers q>5, and the Suzuki groups Sz(e2) and the Ree groups R(e3) for all odd e⩾3. We also show that L2(q) and SL2(q) admit strongly real Beauville structures, yielding real Beauville surfaces, for all q>5.