  Let $\mathcal{P}$ be a countable multiset of primes and let
$G=\bigoplus_{p\in P}\mathbb{Z}/p\mathbb{Z}$. We study the universal
characteristic factors associated with the Gowers-Host-Kra seminorms for the
group $G$. We show that the universal characteristic factor of order $<k+1$ is
a factor of an inverse limit of finite dimensional $k$-step nilpotent
homogeneous spaces. The latter is a counterpart of a $k$-step nilsystem where
the homogeneous group is not necessarily a Lie group. This result provides a
counterpart of the structure theorem of Host-Kra and Ziegler concerning
$\mathbb{Z}$-actions and generalizes the results of Bergelson Tao and Ziegler
concerning $\mathbb{F}_p^\omega$-actions. This result is the first instance of
a structure theorem for the universal characteristic factors associated with a
non-finitely generated group of unbounded torsion. As an application we derive
an alternative proof for the $L^2$-convergence of multiple ergodic averages
associated with $k$-term arithmetic progressions in $G$ and derive a formula
for the limit in the special case where the underlying space is a nilpotent
homogeneous system.
