TITLE: Dessins d'enfants and absolute Galois group
QUESTION [11 upvotes]: I would like to know what is the recent progress about the group homomorphism
$$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$

$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ is the absolute Galois group of $\mathbf{Q}$.
$\mathrm{Out}(\hat{F_{2}})$ is the group of outer automorphisms of the procompletion of the free group generated by two elements. 

It is know to be injective homomorphism, what can we say about the image ?

REPLY [7 votes]: For the sake of getting this off the unanswered stack...
Please refer to the following reference:

Guillot, P. An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group, Enseign. Math. vol 60, 2014. [arXiv version | journal version]

The article just cited develops the theory of dessins d'enfants from scratch. The final section discusses the image of the homomorphism mentioned in the OP's question. 
In particular,

Guillot includes a proof that the image must lie in the group
$\widehat{GT_0}$, a proper subgroup of ${\rm Out}(\hat{F_2})$, first
defined (I believe) by Drinfeld;
Guillot discusses (but does not prove) a stronger result due to
Ihara that the image must lie in the Grothendieck-Teichmuller group,
$\widehat{GT}$, a proper subgroup of $\widehat{GT_0}$.

Please consult the bibliography of the given reference to find the relevant works of Drinfeld and Ihara.