Let $G$ be finite abelian group and $n>2$ integer and $a_1,\ldots,a_n$ distinct elements of $G$.
Let $L$ be a system of equations over $G$ in $n$ variables.
Is it possible the solutions of $L$ to be exactly the permutations of $a_1,\ldots,a_n$?
For $n=2$ it is possible for $G$ the additive subgroup of $\mathbb{F}_2$.