$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Out{Out}$It is true for $g=1,2$. $\Mod(S_1) \cong \GL_2(\mathbb{Z})\cong \Mod(S_{1,1})\cong \Out(F_2)$.

In $\Mod(S_2)$, the hyperelliptic involution is central and fixes the six Weierstrass points on any Riemann surface of genus 2. Since any mapping class group element commutes with this, it will permute the 6 Weierstrass points, and thus one may embed $\Mod(S_2)\leq \Mod(S_{6,2})$, the 6-punctured surface of genus 2 (this follows eg from Birman-Hilden theory, or see the comment on p. 2 of [this paper][1]). Hence $\Mod(S_2) < \Out(F_{9})$.

Notice that we are obtaining a splitting of the homomorphism  
$\Mod(S_{6,2})\to \Mod(S_2)$. I wouldn’t expect such a splitting from $\Mod(S_{k,g})\to \Mod(S_g)$ for $g>2$, and doesn’t exist for $g>3$ by Theorem A of the above linked paper.


  [1]: https://arxiv.org/abs/1804.11235