  Let $\mathcal{P}$ be an (unbounded) countable multiset of primes (i.e. every
prime may appear multiple times), let $G=\bigoplus_{p\in
\mathcal{P}}\mathbb{F}_p$. We develop a Host-Kra structure theory for the
universal characteristic factors of an ergodic $G$-system. More specifically,
we generalize the main results of Bergelson Tao and Ziegler who studied these
factors in the special case $\mathcal{P}=\{p,p,p,...\}$ for some fixed prime
$p$. As an application we deduce a Khintchine-type recurrence theorem in the
flavor of Bergelson Tao and Ziegler and Bergelson Host and Kra.
