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.

Thanks

let $G$ be an locally finite abelian group, and let $H$ be the quotient of $G$ by a finite subgroup. It it true that $H$ is isomorphic to a subgroup of $G$?. The "weaker question" is the same but with the question replaced withIt is true that $H$ has a finite subgroup $N$ such that $H/N$ is isomorphic to a subgroup of $G$?$\endgroup$ – YCor May 19 '17 at 22:01