  A detailed proof is given of a theorem describing the centraliser of a
transitive permutation group, with applications to automorphism groups of
objects in various categories of maps, hypermaps, dessins, polytopes and
covering spaces, where the automorphism group of an object is the centraliser
of its monodromy group. An alternative form of the theorem, valid for finite
objects, is discussed, with counterexamples based on Baumslag--Solitar groups
to show how it fails more generally. The automorphism groups of objects with
primitive monodromy groups are described, as are those of non-connected
objects.
