  We discuss general properties of strong G$_2$-structures with torsion and we
investigate the twisted G$_2$ equation, which represents the G$_2$-analogue of
the twisted Calabi-Yau equation for SU$(n)$-structures introduced by
Garcia-Fern\'andez - Rubio - Shahbazi - Tipler. In particular, we show that
invariant strong G$_2$-structures with torsion do not occur on compact non-flat
solvmanifolds. This implies the non-existence of non-trivial solutions to the
twisted Calabi-Yau equation on compact solvmanifolds of dimensions $4$ and $6$.
More generally, we prove that a compact, connected homogeneous space admitting
invariant strong G$_2$-structures with torsion is diffeomorphic either to $S^3
\times T^4$ or to $S^3 \times S^3 \times S^1$, up to a covering, and that in
both cases solutions to the twisted G$_2$ equation exist. Finally, we discuss
the behavior of the homogeneous Laplacian coflow for strong G$_2$-structures
with torsion on these spaces.
