Let $S$ an orientable surface of any genus, possibly punctured and with boundary: with the classical notation $S=S_{g,n}^{b}$ where $g$ is the genus, $n$ the number of punctures and $b$ the numer of boundaries.
There is a well known morphism $\Psi \colon Homeo(S) \rightarrow Out(\pi_1(S))$ associating to an homeorphism $f$ the equivalence class of the map $\psi_f \in Aut(\pi_1(S))$ taking the class $[\gamma] \in \pi_1(S)$ to $[f(\gamma)]$.
This map descends to a morphism $\Psi \colon Mod(S) \rightarrow Out(\pi_1(S))$ which is more studied in geometric group theory, but now we are interseted in the first.
I need an example where this map $\Psi \colon Homeo(S) \rightarrow Out(\pi_1(S))$ is not surjective. If we considered only orientation perserving homemorphisms then the question would reduce to the study of the mapping class group action on $Out(\pi_1(S))$, but we consider even orientation reversing homeomorphisms.
Thanks in advance for your help.