This question is a follow-up to Classification of (not necessarily connected) compact Lie groups. In the answer to that question, @LSpice proved that any compact, not necessarily connected Lie group $G$ takes the form $$ G = \frac{G_0 \rtimes R}{P} $$ where $G_0$ is the identity component of $G$, $R$ is a finite group, and $P$ is finite and central.

Nonetheless, there are many possibilities for the semi-direct product. To narrow the list, it would be convenient to separate out those elements of $R$ that act by non-trivial outer automorphisms on $G_0$ and modify the rest so that they commute with $G_0$. The simplest possibly-correct realization of this is:

Hypothesis: Any compact Lie group $G$ can be written in the form $$ G = \frac{(G_0 \times H) \rtimes R}{P} $$ where $H, R, P$ are finite groups, non-trivial elements of $R$ act by non-trivial outer automorphisms on $G_0$, and $P$ is central.

Does anyone know if this is true (or can supply counterexamples)? Any idea how it might be proved?