I am trying to prove the following: Suppose that $\alpha:H\to G$ is a Lie group homomorphism and let $P\to M$ be a principal $H$-bundle and $Q\to M$ a principal $G$-bundle. Suppose further that there is a connection 1-form $\vartheta_P$ on $P$. I want to show that if $$ \varphi:P\to Q $$ is an extension of $P$ to $G$, then there is a (unique?) connection $\vartheta_Q$ on $Q$ such that $$ \varphi^*\vartheta_Q = \alpha_*\circ \vartheta_P $$
I am having difficulty trying to define $\vartheta_Q$. Any suggestions are welcome! Thanks!