Classifying certain classes of 2-arc-transitive graphs has received considerable attention. Such classification results have been established for 2-arc-transitive Cayley graphs of abelian groups and dihedral groups. Aiming at a generalization of these results, a project was started by Professor Cai Heng Li and the speaker to classify 2-arc-transitive Cayley graphs of solvable groups. A crucial part of this project is a classification of factorisations of almost simple groups with a solvable factor, which has its own importance in group theory. In this talk, I'll first give a brief introduction to the study of graph symmetries and then sketch the main result and approach in out project.