  Recently, Gareth Jones observed that every finite group $G$ can be realized
as the group of automorphisms of some dessin d'enfant ${\mathcal D}$. In this
paper, complementing Gareth's result, we prove that for every possible action
of $G$ as a group of orientation-preserving homeomorphisms on a closed
orientable surface of genus $g \geq 2$, there is a dessin d'enfant ${\mathcal
D}$ admitting $G$ as its group of automorphisms and realizing the given
topological action. In particular, this asserts that the strong symmetric genus
of $G$ is also the minimum genus action for it to acts as the group of
automorphisms of a dessin d'enfant of genus at least two.
