Suppose you have a group $F$, and a characteristic subgroup $K\le F$. Under what conditions on $F$ and $K$ is the closure $\overline{K}$ inside the profinite completion $\widehat{F}$ also characteristic?
I'm mostly happy to assume that $F$ is finitely generated, though I'd like to know as much about this as possible.
Here's one sufficient condition: If every surjection $F\rightarrow F/K$ is equivalent up to an automorphism of $F/K$, then $\overline{K}$ is characteristic in $\widehat{F}$.
Proof: There's a bijection between $Hom(F,F/K)$ and $Hom(\widehat{F},F/K)$. Thus, by our hypothesis, every surjection $\widehat{F}\rightarrow F/K$ is also equivalent up to $Aut(F/K)$, and hence they all have the same kernel $\overline{K}$, but the intersection of all kernels of surjections to $F/K$ is characteristic, hence $\overline{K}$ is characteristic.
