Let $MCG(S_{g,b}^s)$ be the mapping class group of a surface $S_{g,b}^s$. Assume that it is not trivial.

Is it true that $MCG(S_{g,b}^s)$ is isomorphic to $MCG(S_{g',b'}^{s'})$ if and only if $ S_{g,b}^s$ is homeomorphic to $S_{g',b'}^{s'}$ ? Or what is the complete list of counterexamples?