  Let $G$ be a countable abelian group. We study ergodic averages associated
with arithmetic progressions $\{ag,bg,(a+b)g\}$ for some $a,b\in\mathbb{Z}$.
Under some assumptions on $G$, we prove that the universal characteristic
factor for these averages is a factor of a $2$-step nilpotent homogeneous
space. As an application we derive a Khintchine type recurrence result. In
particular, we prove that for every countable abelian group $G$, if
$a,b\in\mathbb{Z}$ are such that $aG,bG,(b-a)G$ and $(a+b)G$ are of finite
index in $G$, then for every set $E\subset G$ and every $\varepsilon>0$ the
set, $$\{g\in G : d(E\cap E-ag\cap E-bg\cap E-(a+b)g)\geq
d(E)^4-\varepsilon\}$$ is syndetic. This generalizes previous results for
$G=\mathbb{Z}$, $G=\mathbb{F}_p^\omega$ and $G=\bigoplus_{p\in P}\mathbb{F}_p$
by Bergelson Host and Kra, Bergelson Tao and Ziegler and the author,
respectively.
