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I want to expand upon an earlier (partially answered) question: Is a principal bundle of a principal bundle still principal?

As in said question, let $M, P_1, P_2$ be manifolds, let $P_1 \overset{\pi_1}{\to} M$ be a principal bundle corresponding to a Lie group $G_1$, and let $P_2 \overset{\pi_2}{\to} M$ correspond to a Lie group $G_2$. Consider the fibre bundle $P_2 \to M$.

In the original question it was hypothised the composition is a principal bundle with group $G_1 \times G_2$, but this was disproved in the comments by looking at the double cover $S^1 \to S^1 \to pt$ with groups $\mathbb Z/2$ and $S^1$ and noting that the combined map just has group $S^1 \neq \mathbb Z/2 \times S^1$.

However, $1 \to \mathbb Z/2 \to S^1 \to S^1 \to 1$ is exact, so my questions are:

  1. When is $P_2 \to M$ again principal? (Intuitively I expect you might need $G$ is compact, or something like that.)
  2. In that case, is the corresponding group always a semi-direct product of $G_1$ and $G_2$?

(This question was motivated by the physics question of whether it is sensible to apply Yang-Mills theory on the total space of the frame tangent bundle of your physical manifold instead of the manifold itself.)

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