I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components.
With $G_0$ I mean the component connected to the identity which is normal. I would like to ask?
What are the identity components of $Aut(G)$ and $Aut(G_0)$? Let's call them $Aut_0(G)$ and $Aut_0(G_0)$.
Moreover:
Is there a relation between $Aut_0(G)$ (or $Aut_0(G_0)$), and $Inn(G)$ (or $Inn(G_0)$)? Under which hypotheses?
And since they are both normal:
Is there a relation between $Aut(G)/Aut_0(G)$ and $Out(G)$? (Same for $G_0$.)
Last question:
Are there any relations like the ones I asked above, but between the Lie algebras?
Any insight is appreciated, sources and references are very welcome.