The outer automorphism group of a topological group $G$ is constructed by the short exact sequence
$$
1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \operatorname{Out}(G) \longrightarrow 1.
$$
This sequence does not always split, see https://mathoverflow.net/questions/293436/, [for example][1] for the disrete group $G = A_6$.

I am interested in the case where $G$ is a compact, connected Lie group. Does the sequence always split in this case? (If $G$ has a simple Lie algebra $\mathfrak{g}$ then I believe the answer is [yes][2].)


  [1]: https://en.wikipedia.org/wiki/Outer_automorphism_group#In_symmetric_and_alternating_groups
  [2]: https://en.wikipedia.org/wiki/Outer_automorphism_group#In_complex_and_real_simple_Lie_algebras