Classification of a family of symmetric graphs with complete 2-arc-transitive quotients

In this paper we give a classification of a family of symmetric graphs with complete 2-arc-transitive quotients. Of particular interest are two subfamilies of graphs which admit an arc-transitive action of a projective linear group. The graphs in these subfamilies can be defined in terms of the cross ratio of certain 4-tuples of elements of a finite projective line, and thus may be called the second type ‘cross ratio graphs’, which are different from the ‘cross ratio graphs’ studied in A. Gardiner, C. E. Praeger, S. Zhou, Cross-ratio graphs, J. London Math. Soc. (2) 64 (2001), 257–272]. We also give a combinatorial characterisation of such second type cross ratio graphs.