Connection on isomorphic principal bundle

Question

I’m trying to find a natural induced connection $1$-form on the isomorphic principal bundle, more explicitly, given a principal bundle isomorphism $(P:P_1\rightarrow P_2$, $\Phi:G\rightarrow H)$ between principal bundles $G\rightarrow P_1\rightarrow M$ and $H\rightarrow P_2\rightarrow M$ on the same base manifold, and a connection $1$-form $A$ on $P_1$, then how to construct a natural connection $1$-form on $P_2$ by using the isomorphism $P$? I have tried to use the pull back construction, but realized that the induced diffeomorphism from $P$ is trivial, and hence will not work. Any suggestions? Perhaps you can give the correct formula, and I will try to prove it by myself, I just can’t see the construction... Thank you!