  Based on a general formula due to R.Bryant, we work out the topological
structure of the space of torsion-free $G_2$-structures generating the same
associated Riemannian metric on a compact $7$-manifold. We also identify a
corresponding Lie group-theoretic structure of the space. These observations
are then used to describe the moduli space of torsion-free $G_2$-structures in
certain cases - by way of covering spaces.
