  We show that the Grothendieck-Teichm\"uller group of $PSL(2, q)$, or more
precisely the group $GT_1(PSL(2, q))$ as previously defined by the author, is
the product of an elementary abelian 2-group and several copies of the dihedral
group of order 8. Moreover, when $q$ is even, we show that it is trivial.
  We explain how it follows that the moduli field of any "dessin d'enfant"
whose monodromy group is $PSL(2, q)$ has derived length less than 4.
  This paper can serve as an introduction to the general results on the
Grothendieck-Teichm\"uller group of finite groups obtained by the author.
