Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the pure mapping class group of $S$, respectively. Suppose $n >0$ and fix a base point on a boundary. I know that there is a natural monomorphism $$\operatorname{Mod}(S_{g,p}^1)\to \operatorname{Aut}(\pi_1(S_{g,p}^1)),$$ and the image of $\operatorname{Mod}(S_{0,p}^1)$, which is isomorphic to the braid group, $\operatorname{PMod}(S_{0,p}^1)$, which is isomorphic to the pure braid group, and $\operatorname{Mod}(S_{g,0}^1)=\operatorname{PMod}(S_{g,0}^1)$ are known: respectively $$\{\phi \mid \phi(x_i)\textrm{ is a conjugate of }x_{\sigma(i)},\;\sigma\in \mathfrak{S}_p, \; \phi \textrm{ fixes the boundary}\},$$ $$\{\phi \mid \phi(x_i)\textrm{ is a conjugate of }x_i,\; \phi \textrm{ fixes the boundary}\},$$ $$\{\phi \mid \phi \textrm{ fixes the boundary}\},$$ where $x_i$ correspond the punctures, $\mathfrak{S}_p$ is the symmetric group.

How about the image of $\operatorname{PMod}(S_{g,p}^1)$, which is isomorphic to $\operatorname{Mod}(S_{g,0}^{p+1})/\langle\textrm{boundary Dehn twists}\rangle$), and $\operatorname{Mod}(S_{g,p}^1)$?

(Is the former $$\{\phi \mid \phi(x_i)\textrm{ is a conjugate of }x_i,\; \phi \textrm{ fixes the boundary}\}?)$$

At least, I want to know whether they are known or open.