  For each of the 14 classes of edge-transitive maps described by Graver and
Watkins, necessary and sufficient conditions are given for a group to be the
automorphism group of a map, or of an orientable map without boundary, in that
class. Extending earlier results of Siran, Tucker and Watkins, these are used
to determine which symmetric groups $S_n$ can arise in this way for each class.
Similar results are obtained for all finite simple groups, building on work of
Leemans and Liebeck, Nuzhin and others on generating sets for such groups. It
is also shown that each edge-transitive class realises finite groups of every
sufficiently large nilpotence class or derived length, and also realises
uncountably many non-isomorphic infinite groups. Edge-transitive embeddings of
complete graphs are classified, and there is a detailed discussion of
edge-transitive maps with boundary.
