  We study the Laplacian coflow and the modified Laplacian coflow of
$G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian
coflow we show that the solution is always ancient, that is it is defined in
some interval $(-\infty,T)$, with $0<T<+\infty$. However, for the modified
Laplacian coflow, we prove that in some cases the solution is defined only on a
finite interval while in other cases the solution is ancient or eternal, that
is it is defined on $(-\infty, \infty)$.
