It is well known that every subgroup $H$ of a finite abelian group $G$ is isomorphic to a quotient of $G$. I'm wondering whether there is a counterpart for profinite groups.
For example is it true that every open subgroup $H$ of an abelian profinite group $G$ isomorphic to a quotient?
what about the weaker statement that every open subgroup contains an open subgroup which is isomorphic to a quotient?
If G is a direct product of finite groups then the second weaker statement is true because every open subgroup $H$ contains a cylinder neighborhood of the origin $H'$ for which $G=H'\times K$ where $K$ is a finite product of finite groups.