Forgive my ignorance on the Lie theory. I have the following questions in my current work concerning a certain property of compact connected Lie groups.

First, allow me to fix some notations. Let $G$ be a connected compact Lie group, $\mathfrak{g}$ its Lie algebra. Then $\operatorname{Aut}(G)$, the group of (smooth) automorphisms of the $G$, is a closed subgroup of $\operatorname{Aut}(\mathfrak{g})$, hence is a Lie group itself. Let $\operatorname{ad} : G \to \operatorname{Aut}(G) \subseteq \operatorname{Aut}(\mathfrak{g})$ be the adjoint representation and denote its image by $\operatorname{Inn}(G)$, and needless to say, $\operatorname{Inn}(G)$ is a normal subgroup of $\operatorname{Aut}(G)$. We denote the quotient group $\operatorname{Aut}(G) / \operatorname{Inn}(G)$ by $\operatorname{Out}(G)$, and this is what I mean when I say the group of outer automorphisms in the title.

**Question A.** When $\operatorname{Out}(G)$ is countable, is $\operatorname{Inn}(G)$ always open in $\operatorname{Aut}(G)$?

More generally, although this seems to have a higher chance to have a negative answer, one can ask the following question, which I suspect is still true:

**Question B.** Is $\operatorname{Inn}(G)$ always open in $\operatorname{Aut}(G)$? If **Question A** has an affirmative answer, this is equivalent to ask if $\operatorname{Out}(G)$ is always countable.

**Question C.** When $\operatorname{Inn}(G)$ is open in $\operatorname{Aut}(G)$, is $\operatorname{Out}(G)$ (which now a countable discrete group) always amenable as a locally compact group?

A little experiment on the abelian case of the $n$-dimensional torus (in which case the inner automorphism group is trivial, and $\operatorname{Aut}(T^n)$ is $GL_n(\mathbb{Z})$), and the semisimple case (at least over $\mathbb{C}$ instead of $\mathbb{R}$, which implies $\operatorname*{Out}(G)$ being finite), seems to make a negative answer to the above questions not completely trivial. Although I am happy that the abelian case and the semisimple case already suffice for the purpose of my current work, which only needs $\operatorname{Out}(G)$ to be amenable, I would be very interested if one can prove or disprove the amenability of $\operatorname*{Out}(G)$ for a general compact connected Lie group $G$.