Inspired by the question 

http://mathoverflow.net/questions/103120/does-the-moduli-space-of-smooth-curves-of-genus-g-contain-an-elliptic-curve

and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth projective connected) curves of genus $3$ contains a (smooth projective connected) curve of genus $2$. 

The existence of such a genus two curve is equivalent to the existence of a surface $S$, a genus two curve $C$ and a smooth projective morphism $S\to C$ whose fibres are genus three curves.

If the answer is positive, how explicit can our answer be made? I'm already aware of the fact that $M_g$ contains a complete curve for all $g\geq 2$. For instance, in the  paper by Chris Zaal 

http://dare.uva.nl/document/38546

 many curves of some genus (I think 513) are shown to embed into $M_3$.

Of course, by Shafarevich' conjecture, the set $M_3(C)$ is finite for any genus two curve $C$. I'm asking whether it is non-empty for some genus two curve $C$. 

There are many related questions I'd also like to ask. For example, what is the minimal $g$ such that $M_g$ contains a genus two curve? Or, what is the minimal $g$ such that $M_3$ contains a genus $g$ curve? And, finally, is there an example of a complete curve in $M_g$ which is defined over $\overline{\mathbf{Q}}$? But I'll stick to the above question for now.