Let $S=S\_{g,b}$ be a compact orientable surface with genus $g$ and $b$ boundary components, such that $\chi(S)=2-2g-b<0$. Let $Q=\{x\_1,\ldots , x\_n\}$ be a set of $n$ distinguished points in the interior of $S$.


Define ${\rm Mod}(S,\{Q\})$ to be the mapping class group of orientation-preserving self-homeomorphisms of $S$ which fix the boundary $\partial S$ pointwise and preserve $Q$ setwise, modulo isotopies of the same type. I believe this is the same as the mapping class group ${\rm Mod}(S-Q)$ of the punctured surface with boundary $S-Q$.


Now fixing a basepoint $d\in \partial S$, there are homomorphisms
$$
{\rm Mod}(S,\{Q\})\to {\rm Aut}(\pi\_1(S-Q,d))\to {\rm Out}(\pi\_1(S-Q,d)). 
$$



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> Is there a version of the Dehn-Nielsen-Baer theorem which states that the above composition is isomorphic onto the subgroup ${\rm Out}^\*(\pi\_1(S-Q,d))$ of ${\rm Out}(\pi\_1(S-Q,d))$ consisting of outer automorphisms which preserve the individual conjugacy classes represented by the components of $\partial S$, and permute the conjugacy classes represented by simple closed curves around the punctures?
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We have looked in the following sources, which are all very helpful but seem to only treat the cases of boundary and punctures separately:


*Farb, Benson; Margalit, Dan*, A primer on mapping class groups, Princeton Mathematical Series. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/ebook). xiv, 492 p. (2011). [ZBL1245.57002](https://zbmath.org/?q=an:1245.57002).


*Ivanov, Nikolai V.*, Mapping class groups, Daverman, R. J. (ed.) et al., Handbook of geometric topology. Amsterdam: Elsevier. 523-633 (2002). [ZBL1002.57001](https://zbmath.org/?q=an:1002.57001).


*Zieschang, Heiner; Vogt, Elmar; Coldewey, Hans-Dieter*, Surfaces and planar discontinuous groups. (Poverkhnosti i razryvnye gruppy)., Moskva: Nauka. 688 p. (1988). [ZBL0701.57001](https://zbmath.org/?q=an:0701.57001).


*Boldsen, Soren,* Different versions of mapping class groups of surfaces, <https://arxiv.org/abs/0908.2221>



Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures
The issue here is Dehn twists along curves parallel to the circles of $\partial S$. These usually generate infinite cyclic subgroups of ${\rm Mod}(S,Q)$, the only exceptions being when $S$ is a disk and $Q$ is empty or a single point. If one chooses a basepoint in $\partial S$ then these Dehn twists induce the identity on $\pi\_1(S-Q)$ except for a twist along a curve parallel to the component of $\partial S$ containing the basepoint, which induces the inner automorphism which ccnjugates $\pi\_1(S-Q)$ by the loop given by this component of $\partial S$. Thus the composition ${\rm Mod}\to {\rm Aut}\to {\rm Out}$ will usually have a nontrivial kernel. 


