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 first stronger statement is correct since there exists a splitting $G=H\times K$ (both topological and algebraic) for a finite subgroup $K$.

Thanks.