Let $P$ be a principal $G$ bundle. Let $S$ be a space with left action of $G$, and let $Q$ be a principal $H$ bundle over $S$ with the property that the action of $G$ can be lifted to $Q$.

Then $$ P \times_G Q $$ is a principal $H$ bundle over $$ P \times_G S $$ with projection map $\pi([p,q])=[p,\pi_Q(q)]$, where $\pi_Q:Q \rightarrow S$ is the projection of the principal bundle $Q$. One checks that $\pi$ is well-defined, i.e. $\pi([pg^{-1},gq])=[pg^{-1},g \pi_Q(q)]$, which holds because the $G$ action on $Q$ is lifted from $S$.

What ingredients do I need to get an induced connection on the principal $H$ bundle $P \times_G Q \rightarrow P \times_G S$?

I hope the answer to be similar to the ordinary case of connections on associated bundles, where every principal bundle connection induces a connection on the associated bundle. I played around with a principal bundle connection on $P$, and a principal bundle connection that is $G$-invariant on $Q$, but could not write down anything on the new bundle.