Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $Mod(S)$ and $PMod(S)$ 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 $$Mod(S_{g,p}^1)\to Aut(\pi_1(S_{g,p}^1))$$, and the image of $Mod(S_{0,p}^1)$ (which is isomorphic to the braid group), $PMod(S_{0,p}^1)$ (which is isomorphic to the pure braid group), $Mod(S_{g,0}^1)=PMod(S_{g,0}^1)$ are known: $\{\phi | \phi(x_i)\textrm{ is a conjugate of }x_{\sigma(i)}, \sigma\in \mathfrak{S}_p, \phi \textrm{ fixes the boundary}\}$, $\{\phi | \phi(x_i)\textrm{ is a conjugate of }x_i, \phi \textrm{ fixes the boundary}\}$, $\{\phi | \phi \textrm{ fixes the boundary}\}$, respectively, where $x_i$ correspond the punctures, $\mathfrak{S}_p$ is the symmetric group.

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