Poisson approximation for the distribution of the number of “parallelograms” in a random sample from $\mathbb Z_N^q$

For a random sample with replacement $\xi_1,\dots,\xi_T$ from a group $\mathbb Z_N^q$, $N\geq4$, we consider the distribution of the number $\zeta$ of 4-element subsets satisfying the relation of type $\xi_{i_1}-\xi_{i_2}=\xi_{i_3}-\xi_{i_4}$ and additional condition given in terms of a metric on this group. Estimates of the accuracy of Poisson approximation for the distribution of $\zeta$ are obtained and conditions of the weak convergence of $\zeta$ to the Poisson law are established.