  We study double averages along orbits for measure preserving actions of
$\mathbb{A}^\omega$, the direct sum of countably many copies of a finite
abelian group $\mathbb{A}$. In this article we show an $L^p$ norm-variation
estimate for these averages, which in particular reproves their convergence in
$L^p$ for any finite $p$ and for any choice of two $L^\infty$ functions. The
result is motivated by recent questions on quantifying convergence of multiple
ergodic averages.
