  We consider left-invariant (purely) coclosed G$_2$-structures on
7-dimensional 2-step nilpotent Lie groups. According to the dimension of the
commutator subgroup, we obtain various criteria characterizing the Riemannian
metrics induced by left-invariant purely coclosed G$_2$-structures. Then, we
use them to determine the isomorphism classes of 2-step nilpotent Lie algebras
admitting such type of structures. As an intermediate step, we show that every
metric on a 2-step nilpotent Lie algebra admitting coclosed G$_2$-structures is
induced by one of them. Finally, we use our results to give the explicit
description of the metrics induced by purely coclosed G$_2$-structures on
2-step nilpotent Lie algebras with derived algebra of dimension at most two, up
to automorphism.
