The resolution of the G2-orbifold T^7/Γ, where Γ is a suitably chosen finite group, admits a 1-parameter family of G2-structures with small torsion φ^t, obtained by gluing in Eguchi-Hanson spaces. It was shown by Joyce that φ^t can be perturbed to a torsion-free G2-structure \tilde{φ}^t for small values of t. Using norms adapted to the geometry of the manifold we give an alternative proof of the existence of \tilde{φ}^t. This alternative proof produces the estimate || \tilde{φ}^t-φ^t ||_{C^0} ≤ ct^{5/2}. This is an improvement over the previously known estimate || \tilde{φ}^t-φ^t ||_{C^0} ≤ ct^{1/2}. As part of the proof, we show that Eguchi-Hanson space admits a unique (up to scaling) harmonic form with decay, which is a result of independent interest.

More generally, there exists a construction of torsion-free G2-structures on resolutions of a more general class of G2-orbifolds, given by Joyce and Karigiannis. We explain a construction of G2-instantons on these manifolds, which includes the case of G2-instantons on resolutions of T^7/Γ as a special case. The ingredients needed are a G2-instanton on the orbifold and a Fueter section over the singular set of the orbifold. In the general case, we make the very restrictive assumption that the Fueter section is pointwise rigid. In the special case of resolutions of T^7/Γ, the improved estimate for \tilde{φ}^t-φ^t allows to remove this assumption. As an application, we construct one new example of a G2-instanton on the resolution of (T^3 × K3)/\Z^2_2.