For a surface $S$, we denote the mapping class group of $S$ as $\operatorname{Map}(S)$.
Question: Are there surfaces $S$ such that there exists some subgroup $H<\operatorname{Map}(S)$ such that $H\cong\operatorname{Map}(S)$?
Follow up: If so, are there conditions we can place on $S$ to always guarantee such an $H$?
This question has a small partial answer in the affirmative, namely when a surface has a proper subsurface homeomorphic to itself. However, this requires for $S$ to have boundary. Moreover, all examples I have come up with are of infinite-type. What can we say in the case when $S$ has no boundary, or in the finite-type case?
Edit: The paper On injective homomorphisms between Teichmuller modular groups gives a negative answer when $S$ is a closed orientable surface of genus greater than two.