I'm not sure of the difficulty of the question I'm about to ask. If it does not fit the criteria for this site then I apologize in advance, I'm rather new here.
So here it goes: Let $G$ be a Lie group and $Mod(G)$ be the mapping class group of the underlying manifold (by mapping class group I mean homeomorphisms from $G$ to itself up to isotopy). Denote the group of Lie-group isomorphisms as $Aut(G)$.
In general, what is the relationship between $Mod(G)$ and $Aut(G)$?
It is clear that $Aut(G)$ injects the into $Homeo(G)$ (the set of all homeomorphisms of $G$ to itself) and we know $Mod(G)\cong Homeo(G)/homotopy$. However, there are certainly homeomorphisms in $Homeo(G)$ that do not preserve the group structure and moreover, an element of $Aut(G)$ must fix the identity. I guess a specific question of importance would be:
When are two elements of $Aut(G)$ isotopic as homeomorphisms?
Also, would the case change if we were to work over a different category for $Mod(G)$ (i.e. PL, Diffeo, ect.) I hope this all makes sense and meets the standards of the board.