A short exact sequence with a locally finite group 
If your group $G$ belongs to a short exact sequence $$1 \to F \to G \to \mathbb{Z}^2 \to 1,$$ where $F$ is a locally finite group, is it true that $G$ necessarily contains a copy of $\mathbb{Z}^2$?

If $F$ is finite, this is clearly true: If $a,b \in G$ lift a basis of $\mathbb{Z}^2$, then $[a,b^k] \in F$ for every $k \geq 1$, hence $[a,b^p]=[a,b^{p+q}]$ for some $p,q \geq 1$. We conclude that $[a,b^q]=1$, so that $\langle a,b^q \rangle$ defines a copy of $\mathbb{Z}^2$ in $G$.
Is it still true when $F$ is locally finite?
EDIT: A group is locally finite if its finitely-generated subgroups are finite.
 A: The answer turns out to be negative: Houghton's group $H_3$ is (locally finite)-by-$\mathbb{Z}^2$ but it does not contain $\mathbb{Z}^2$.
The $n$-th Houghton's group is defined as follows. Let $R_n$ denote the disjoint union of $n$ rays (ie., graphs whose vertices are the integers and whose edges link two consecutive integers). Then $H_n$ is the group of the bijections $R_n^{(0)} \to R_n^{(0)}$ which preserve the ends at infinity of $R_n$ and which preserve adjacency for all but finitely many pairs of vertices. 
Any element of $H_n$ defines a translation on each ray (if you look at sufficiently far away), hence a morphism $H_n \to \mathbb{Z}^n$. However, in order to get a bijection, the sum of all the translation lengths must be zero. So the previous morphism provides a surjective morphism $H_n \to \mathbb{Z}^{n-1}$. The kernel of this morphism is included into the commutator subgroup of $H_n$, which coincides with the collection of the elements of $H_n$ which fix each ray sufficiently far away from their origins. So the commutator subgroup is isomorphic to $S_{\infty}$, the union of the symmetric groups $\bigcup\limits_{n \geq 1} S_n$. Consequently, we have the short exact sequence $$1 \to S_\infty \to H_n \to \mathbb{Z}^{n-1} \to 1.$$
The fact that $H_3$ does not contain $\mathbb{Z}^2$ is not obvious. It follows for instance from the article Centralisers in Houghton's groups, written by Simon St John-Green, which shows that the centraliser of an infinite-order element of $H_3$ must be virtually cyclic. 
More generally, the greatest rank of a free abelian subgroup of $H_n$ is $\lfloor n/2 \rfloor$.
