Lifting of continuous group homomorphism

Question

Let $P$, $G$, and $H$ be Lie groups along with a continuous surjective homomorphism $\pi: G\to H$. Suppose there exists a continuous homomorphism $f: P\to H$. I want to know, under what conditions, $f$ can be lifted to a map $s: P \to G$ such that $\pi*s=f$. Of course, if there is an injection $m: H \to G$ then we already have a lift. But is it a necessary condition? I don't understand how to approach the problem. If someone can give some hints that will be very helpful.