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.