I am considering the following situation.

Let $M_5$ be a 5-dimensional manifold which is an $S^1$ principal bundle over 4-manifold $M_4$. For instance, $M_5 = S^5$ and $M_4 = \mathbb{CP}^2$ with standard Hopf fibration.Now there is a principal G-bundle (or associated bundle $adP$) over $M_5$.

**My question is, how to characterize $P$ over $M_5$, and is there a canonical induced bundle over $M_4$?**

Let me explain my question a little bit.

I know that, for even dimensional manifold $X$, characteristic numbers (integrating over $X$ the characteristic class built up by curvature) can efficiently characterize principal. But in odd dimensional cases, I'm not sure how to do similar thing. One thing I can imagine is to integrate characteristic classes over all even-dimensional cycles.

Secondly, suppose there is a way to characterize $P\to M_5$. Then I would like to know if there is a natural principal induced bundle $P'$ over $M_4$, and how to use the data of $P\to M_5$ to describe the induced bundle $P' \to M_4$.

If possible, we can take some simple cases. Say, the Hopf fibration. Principal $G$-bundles over $S^5$ are characterized by $\pi_4G$, and they are $0$ or $\mathbb{Z}_2$, as I check a list of homotopy group for Lie groups. If I take trivial $G$-bundle over $S^5$, do I get a trivial bundle over $\mathbb{CP}^2$ (do I get new "twist" in the process of inducing bundle)? And what about other situation?

Thanks.