We know that if $f:G\rightarrow H$ is a Lie group homomorphism, then the induced map $f_*:\mathfrak{g}\rightarrow\mathfrak{h}$ of their Lie algebras is an algebra homomorphism. It follows relatively easily that if $f$ is furthermore an isomorphism, then so is $f_*$, although the converse does not hold - for instance, if $f:\mathbb{R}\rightarrow\mathbb{S}^1=\mathbb{R}/\mathbb{Z}$ is given by $a\mapsto a+\mathbb{Z}$, then this is not an isomorphism even though the Lie algebras of both sides are isomorphic (to $\mathbb{R}$).
But can we say anything in this reversed direction? In particular I am trying to prove that if $f_*$ is surjective then so is $f$ - I have made almost no progress proving this to be true but have been unable to find a counterexample. Any ideas?
Edit: forgot to mention I am considering all this under the assumption that both $G$ and $H$ are connected (although a comment below has made me realise perhaps only the latter is important).
This holds if $H$ is connected. The fact that $f_{*}$ is onto guarantees that $f$ is surjective onto a neighborhood of the identity of $H$ and since a neighborhood of the identity of $H$ generates $H$ as a group and $f$ is a group homomorphism, we see that $f$ is surjective onto $H$.