Let $G$ be a connected Lie Group. We will denote the outer automorphism group of $G$ by $Out(G)$. Consider $\pi\_0(Diffeo(G))$, the mapping class group of the underlying manifold of $G$, and the map,


$k:Out(G)\rightarrow \pi\_0(Diffeo(G))$


This map is well defined as follows: Given $\phi\_1, \phi\_2\in Aut(G)$ such that $\phi\_1=g\phi\_2g^{-1}$, we may construct a path $\alpha: [0,1]\rightarrow G$ such that $\alpha(0)=g$ and $\alpha(1)=e$, the identity in $G$. Then $H(w,t)=\alpha(t)\phi\_2(w)\alpha(t)^{-1}$ is an isotopy with $H(w,0)=g\phi\_2g^{-1}=\phi\_1$ and $H(w,1)=\phi\_2$.


Consider the case where $G=T^2$, the $2$-torus. Then $Out(T^2)\cong GL\_2(\mathbb{Z})$ and $\pi\_0(Diffeo(T^2))\cong Out(\pi\_1(T^2))\cong GL\_2(\mathbb{Z})$ via the Dehn-Nielsen-Baer Theorem. In this case, $k$ is an isomorphism. 


Another case is the $3$-sphere, in which $Out(S^3)\cong 0$ and $\pi\_0(Diffeo(S^3))\cong\mathbb{Z}\_2$. In this case, $k$ is injective.


Is there a case where $k$ is surjective? If not, what is the obstruction? Moreover, are there higher dimensional examples of the bijective case? It seems that for higher dimensions, the smooth structures may make this problem computationally intractable.



Outer Automorphisms and the Mapping Class Group of Lie Groups
Your question is largely intractable with present technology. Little is known about smooth mapping class groups of high-dimensional manifolds. But in dimensions 3 and below, there is considerable knowledge. . . unfortunately there are not many Lie groups in those dimensions. 


The mapping class group of $SO\_3$ is trivial. I am not certain of who proved it originally, but the uniqueness of the genus 1 Heegaard splitting would be one way to prove it. This approach would be due to Bonahon. 


So for $SO\_3$ you have surjectivity. 


I believe there are no other Lie Groups where there any kind of upper bound on the elements of $\pi\_0 Diff(G)$ known, so to get any results of the kind you are interested in would require some substantial work. 


