The absolute Galois group of the rationals has certain faithful actions through which we may hope to discover some of its structure. Two important examples are its action on the arithmetic fundamental group of P^1 minus three points and its action on dessins d'enfants (i.e. unramified covers of P^1 minus three points). These examples are two sides of the same coin. One goal is to understand something about the absolute Galois group by finding relations on its action on this fundamental group and/or invariants of its action on dessins, and we discuss a method of "Belyi-extending maps" which gives such relations and invariants.