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 $$ {\rm Mod}(S,\{Q\})\to {\rm Aut}(\pi_1(S-Q,d))\to {\rm Out}(\pi_1(S-Q,d)). $$

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?

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.

*Ivanov, Nikolai V.*, Mapping class groups, Daverman, R. J. (ed.) et al., Handbook of geometric topology. Amsterdam: Elsevier. 523-633 (2002). ZBL1002.57001.

*Zieschang, Heiner; Vogt, Elmar; Coldewey, Hans-Dieter*, Surfaces and planar discontinuous groups. (Poverkhnosti i razryvnye gruppy)., Moskva: Nauka. 688 p. (1988). ZBL0701.57001.

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