Let $G$ be a finitely generated group. Suppose we have two families $F_1$ and $F_2$ of finite index subgroups of $G$, and each family has trivial intersection and is filtered from below (i.e. for any two elements in the family their intersection contains some third element).
These families generate two profinite topologies on $G$. (Taking the subgroups in the families as basis of open neighborhoods around identity).
Suppose the completions wrt to these families produce isomorphic profinite groups.
Can we say that these families generate the same topology on $G$?
(Equivalently, given any $N \in F_1$ is there $N_2 \in F_2$ such that $N_1 \leq N_2$ and vice versa.)
What if one family is a subfamily of the other?