Profinite groups and extension of a monomorphism

Question

Let $G$ be a finite group and $G^\vee$ be its profinite completion. Suppose we have an injective homomorphism $\varphi: G \rightarrow H$ where $H$ is a profinite group. By the universal property of profinite completion, $\varphi$ induces a continuous map $\varphi^\vee: G^\vee \rightarrow H.$
My question is:
When is $\varphi$ a monomorphism?

Answer 1

The profinite completion of any profinite topological group is just itself: the identity map $G\to G$ satisfies the universal property of the profinite completion. So $G=G^\vee$ and $\varphi^\vee=\varphi$, so $\varphi^\vee$ is injective if $\varphi$ is injective.