= Solution
The <surface group> of a closed oriented genus-two <surface> $Y$ has presentation
$$
\pi_1(Y)=\langle a_1,b_1,a_2,b_2\mid[a_1,b_1][a_2,b_2]=1\rangle.
$$
Send $a_1\mapsto(12)$, $a_2\mapsto(123)$ and $b_1,b_2\mapsto1$. The relation holds and the images generate $S_3$, giving a surjective <group homomorphism>. Its normal kernel defines a connected six-sheeted regular <covering space> $\widetilde Y\to Y$, with <deck transformation group> $S_3$.
The covering surface is closed and inherits an <orientation>. Multiplicativity of the <Euler characteristic> gives $\chi(\widetilde Y)=6\chi(Y)=6(-2)=-12$, so $2-2g=-12$ and $\boxed{g=7}$. A nonidentity <deck transformation> of a connected cover fixes no point: if it fixed one, uniqueness of path lifts would force it to be the identity everywhere. Thus this supplies the requested \b[free $S_3$ action]. It is a case of <finite groups act freely on suitable closed orientable surfaces>.
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