Lie algebras homomorphism induces Lie groups homomorphism

Question

Given a homomorphism between two Lie algebras $\varphi:\mathfrak g\rightarrow \mathfrak h$. Let $G$ and $H$ be Lie groups with Lie algebras $\mathfrak g$ and $\mathfrak h$ resp. in which cases do we have a corresponding Lie group homomorphism $F:G\rightarrow H$ such that $F$ and $\varphi$ commute with exponential maps?

Answer 1

In general no, but if $G$ is simply connected yes. This is called Lie's Second Theorem.