I am following Kobayashi and Nomizu's book Foundations of differential geometry volume 1 (page no $57$) Proposition 5.6.
The structure group of a principal bundle $P(M,G)$ is reducible to a closed subgroup $H$ of $G$ if and only if the associated bundle $E=(P\times G/H)/G\rightarrow M$ has a global section. (Here we use the notation $Q(M,H)$ for the reduced bundle.)
One direction I was able to understand. Given that $G$ is reducible to $H$, I was able to produce a global section for $E\rightarrow M$.
For the other direction, I was able to understand everything except that I could not prove $Q$ is an immersed submanifold.
If the following result is true, then I am done.
Is the inverse image of an immersed submanifold an immersed submanifold under smooth submersion?
I know the above result is true for embedded submanifold (Using transversality) but I am not sure about the result for immersed submanifold.
Yes - use that an immersion is locally an embedding to reduce to the case of an embedded submanifold.
Added:
Let $U$ be the immersed submanifold with preimage $V$. Cover $U$ by embedded submanifolds $U_i$. The preimages $V_i$ are embedded and cover $V$, so $V$ is an immersion.