Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$ (with $2g+n-2 > 0$), $\Pi_{g,n}$ its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, orientation preserving mapping class group. Then by the Dehn-Nielsen-Baer theorem, the outer action of $\Gamma_{g,n}$ on $\Pi_{g,n}$ is faithful and induces an isomorphism between $\Gamma_{g,n}$ and the subgroup $\text{Out}^*(\Pi_{g,n})$ of $\text{Out}(\Pi_{g,n})$ which fixes the conjugacy class of each simple closed curve surrounding a puncture.

When does $\text{Out}^*(\Pi_{g,n})$ have finite index inside $\text{Out}(\Pi_{g,n})$?

If $(g,n)$ is $(1,1)$, then the index is 2. Since $\Pi_{1,1} = \Pi_{0,3}$ and $\Gamma_{0,3}$ is trivial, you get infinite index for $(0,3)$. What happens in the other cases? I'm particularly interested in the case where $g\ge 2, n = 1$.

Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$, with $2g+n-2 > 0$. Let $\Pi_{g,n}$ be its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, orientation preserving mapping class group. 

Then, by the Dehn-Nielsen-Baer theorem, the outer action of $\Gamma_{g,n}$ on $\Pi_{g,n}$ is faithful and induces an isomorphism between $\Gamma_{g,n}$ and the subgroup $\text{Out}^*(\Pi_{g,n})$ of $\text{Out}(\Pi_{g,n})$ which fixes the conjugacy class of each simple closed curve surrounding a puncture.

Question. When does $\text{Out}^*(\Pi_{g,n})$ have finite index inside $\text{Out}(\Pi_{g,n})$?

If $(g,\,n)=(1,\,1)$, then the index is $2$. Since $\Pi_{1,1} = \Pi_{0,3}$ and $\Gamma_{0,3}$ is trivial, one gets infinite index for $(g, \, n)=(0,\,3)$. What happens in the other cases? I'm particularly interested in the case where $g\ge 2, \, n = 1$.

Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$ (with $2g+n-2 > 0$), $\Pi_{g,n}$ its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, orientation preserving mapping class group. Then by the Dehn-Nielsen-Baer theorem, the outer action of $\Gamma_{g,n}$ on $\Pi_{g,n}$ is faithful and induces an isomorphism between $\Gamma_{g,n}$ and the subgroup $\text{Out}^*(\Pi_{g,n})$ of $\text{Out}(\Pi_{g,n})$ which fixes the conjugacy class of each simple closed curve surrounding a puncture.

When does $\text{Out}^*(\Pi_{g,n})$ have finite index inside $\text{Out}(\Pi_{g,n})$?

You certainly get finite index if $(g,n)$ is $(1,1)$ (index is 2). Since $\Pi_{1,1} = \Pi_{0,3}$ and $\Gamma_{0,3}$ is trivial, you get infinite index for $(0,3)$. What happens in the other cases? I'm particularly interested in the case where $g\ge 2, n = 1$.

Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$ (with $2g+n-2 > 0$), $\Pi_{g,n}$ its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, orientation preserving mapping class group. Then by the Dehn-Nielsen-Baer theorem, the outer action of $\Gamma_{g,n}$ on $\Pi_{g,n}$ is faithful and induces an isomorphism between $\Gamma_{g,n}$ and the subgroup $\text{Out}^*(\Pi_{g,n})$ of $\text{Out}(\Pi_{g,n})$ which fixes the conjugacy class of each simple closed curve surrounding a puncture.

When does $\text{Out}^*(\Pi_{g,n})$ have finite index inside $\text{Out}(\Pi_{g,n})$?

If $(g,n)$ is $(1,1)$, then the index is 2. Since $\Pi_{1,1} = \Pi_{0,3}$ and $\Gamma_{0,3}$ is trivial, you get infinite index for $(0,3)$. What happens in the other cases? I'm particularly interested in the case where $g\ge 2, n = 1$.