Is there a group $G$ with the following properties?
- Finitely presented
- Not virtually solvable
- LERF (that is, every finitely generated subgroup is closed in the profinite topology on $G$).
If this is hard, I can relax requirement 4: Instead of LERF, it's enough that every finitely generated subgroup which has finitely many conjugates (that is, normalizer of finite index in $G$) is closed in the profinite topology on $G$.