# Lie Automorphisms and Isotopy

Let $X$ be a Lie group, $Aut(X)$ be the Lie automorphism group of $X$ (group automorphisms which are also diffeomorphisms), and $Homeo(X)$ be the homeomorphism group of the underlying manifold. For any $f\in Homeo(X)$, does there exist some $g\in Aut(X)$ such that $f$ and $g$ are isotopic?

I hesitate to post this on this board as I realize that most questions are of a very high standard; however, after consulting other sources I can't seem to find any similar results.

Obviously not. If $G=X$ is not connected, then the connected components are the cosets of $G^0$ and therefore all homeomorphic. In particular any permutation of these components can be realized by an homeomorphism of $G$ (and isotypic homeomorphism induce the same permutation), while every Lie group automorphism induces a group automorphism of the quotient $G/G^0$. Hence every disconnected Lie group (e.g. $O(n)$ or every finite group) is a counterexample.
Every automorphism of the simply conected cover $G$ of $SO(3)$ preserves orientation, so no orientation reversing diffeo of $G$, which is a sphere, is isotopic to an automorphism.