The Grothendieck-Teichm\"uller group of $PSL(2,q)$

We show that the Grothendieck-Teichm"uller group of $PSL(2,q)$, or more precisely the group $G{T}_1(PSL(2,q))$ 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.