Let $\left(P_1,\pi_1,M,G_1\right)$ and $\left(P_2,\pi_2,P_1,G_2\right)$ be two principal bundles, where $M$, $P_1$ and $P_2$ are differential manifolds, and $G_1$ and $G_2$ are Lie groups. With these settings, the chain $$ P_2\xrightarrow{\pi_2}P_1\xrightarrow{\pi_1}M $$ would induce $$ \pi_1\circ\pi_2:P_2\to M. $$
My questions are:
(1) Is this $\pi_1\circ\pi_2$ always a principal bundle, and
(2) In the case where $\pi_1\circ\pi_2$ is a principal bundle, is its structure group always $G_1\times G_2$?
Motivation. That $\pi_1\circ\pi_2$ is always a fiber bundle seems straightforward, but I am curious as to whether this hierarchical system preserves the Lie group structure (especially if it does in the direct-product fashion).
Thank you!