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.


Edit: (This is wrong) Ok I believe i can prove the second weaker statement. Let $G$ be a profinite group, then $G$ embeds into a direct product of finite groups $\Gamma$. Let $H$ be an open subgroup of $G$, therefore it is of finite index. So there exists an open subgroup $U\leq \Gamma$ so that $U\cap G\subseteq H$ (Here's [a link](https://math.stackexchange.com/questions/761774/existence-of-particular-open-subgroups-given-a-profinite-group?rq=1)!). Now $U$ contains an open subgroup $W$ so that $\Gamma =W\times K$ Consider the set $W\cap G$ we have a map $\phi:G\rightarrow \Gamma\rightarrow \Gamma/K=W$ This map is onto $W\cap G$ ***(it is onto the set of all $w\in W$ so that there exists $k\in K$ so that $(w,k)\in G$)*** It follows that $W\cap G$ is a subgroup of $H$ that is isomorphic to a quotient.

Please write a comment if you find a mistake.