Let $F$ be a free group of finite rank, and $K\subset F$ a finite index characteristic subgroup.

Let $\hat{F}$ be the profinite completion of $F$ (i.e. a free profinite group of same rank), and $\bar{K}$ the closure of $K$ in $\hat{F}$.

What are necessary and/or sufficient conditions for $\bar{K}$ to be characteristic in $\hat{F}$?

Is there an example of such $K$ for which $\bar{K}$ is not characteristic?

**Motivation:** I'm trying to answer the question whether there exists a finite
(non-abelian) *simple* characteristic quotient of $F$. This will be possible only with a characteristic subgroup whose profinite closure is *not* characteristic.

This question was asked here in a more general setting. A sufficient condition was given in the answer, namely that $K$ be "hyper-characteristic" (or "isomorph-free"). However this doesn't help me, as the quotient by such a subgroup cannot be a non-abeian simple group, so I'm looking for other conditions.

"(i.e., a free profinite group of the same rank)"The profinite completion is not just a profinite group, it's a profinite group along with a homomorphism from the original group. $\endgroup$