  The regular objects in various categories, such as maps, hypermaps or
covering spaces, can be identified with the normal subgroups N of a given group
\Gamma, with quotient group isomorphic to \Gamma/N. It is shown how to
enumerate such objects with a given finite automorphism group G, how to
represent them all as quotients of a single regular object U(G), and how they
are acted on by the outer automorphism group of \Gamma. Examples constructed
include kaleidoscopic maps with trinity symmetry.
