  In the present paper, we show that many combinatorial and topological
objects, such as maps, hypermaps, three-dimensional pavings, constellations and
branched coverings of the two--sphere admit any given finite automorphism
group. This enhances the already known results by Frucht, Cori -- Mach\`i,
\v{S}ir\'{a}\v{n} -- \v{S}koviera, and other authors. We also provide a more
universal technique for showing that ``any finite automorphism group is
possible'', that is applicable to wider classes or, in contrast, to more
particular sub-classes of said combinatorial and geometric objects. Finally, we
show that any given finite automorphism group can be realised by sufficiently
many non-isomorphic such entities (super-exponentially many with respect to a
certain combinatorial complexity measure).
