Given an inclusion $P\subset \Gamma$, it induces a homomorphism $\phi:\hat P\to \hat\Gamma$ on profinite completions. Then is that true that $\phi$ is injective iff given any normal subgroup of finite index $R \subset P$, there exists a subgroup of finite index $S \subset \Gamma$ such that $S\cap P \subset R$?
The question is from a claim in the proof of Theorem 5.1 of this paper. It seems that we need to pass $S$ to its normal core, which is also a finite index subgroup of $\Gamma$, s.t. the intersection with $P$ is contained in $R$. However, I am not sure how to use this since there is no homomorphism from $P/R$ to $\Gamma/S$.