A while back, J. Ellenberg brought the following problem to my attention.

If $G$ is a residually finite group, let $\widehat G$ be its profinite completion. Let $S$ be a closed surface of genus $g \geq 2$, and let $\pi$ be its topological fundamental group. Let $\mathrm{Mod}(S)$ be the mapping class group of $S$.

There is homomorphism $\widehat{\mathrm{Mod}(S)} \to \mathrm{Out}(\widehat \pi)$. Is this map surjective?

In other words, does the geometric fundamental group of the moduli space surject the outer automorphisms of the geometric fundamental group of the curve?

Edit: As Jordan points out, the map is not surjective. So, the question is:

What's the closure of the image of $\mathrm{Mod}(S)$ in $\mathrm{Out}(\widehat \pi)$?

Or, more precisely, for Henry:

As Jordan explains, there is a map $\mathrm{Out}(\widehat \pi) \to \mathrm{Sp}_{2g}(\widehat{\mathbb{Z}}) \to \widehat{\mathbb{Z}}^\star$

Is the closure of the image of $\mathrm{Mod}(S)$ in $\mathrm{Out}(\widehat \pi)$ the preimage of 1 and -1?