Torsion-free $G_2$-structures with identical Riemannian metric

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.