  We study the absolute Galois group by looking for invariants and orbits of
its faithful action on Grothendieck's dessins d'enfants. We define a class of
functions called Belyi-extending maps, which we use to construct new Galois
invariants of dessins from previously known invariants. Belyi-extending maps
are the source of the ``new-type'' relations on the injection of the absolute
Galois group into the Grothendieck-Teichmuller group. We make explicit how to
get from a general Belyi-extending map to formula for its associated invariant
which can be implemented in a computer algebra package. We give an example of a
new invariant differing on two dessins which have the same values for the other
readily computable invariants.
