Closed G₂ structures on compact manifolds satisfying the known topological obstructions to holonomy G₂ metrics

We construct new compact manifolds endowed with closed G₂ structures that satisfy the topological properties found by Joyce and Baraglia for the existence of a torsion-free G₂ structure in the same cohomology class. Those manifolds arise as resolutions of orbifolds of the form M/Z₂, where M itself does not admit any torsion-free G₂ structure. We develop an equivariant resolution procedure for closed G₂ orbifolds with Z₂ isotropy, and analyze some topological properties of the resolved manifolds, including their first Pontryagin class and certain triple Massey products.