  In 1979, Miller proved that for a group $G$ of odd order, two minimal group
codes in $\mathbb{F}_2G$ are $G$-equivalent if and only they have identical
weight distribution. In 2014, Ferraz-Guerreiro-Polcino Milies disprove Miller's
result by giving an example of two non-$G$-equivalent minimal codes with
identical weight distribution. In this paper, we give a characterization of
finite abelian groups so that over a specific set of group codes, equality of
important parameters of two codes implies the $G$-equivalence of these two
codes. As a corollary, we prove that two minimal codes with the same weight
distribution are $G$-equivalent if and only if for each prime divisor $p$ of
$|G|$, the Sylow $p$-subgroup of $G$ is homocyclic.
