Closed $\mathrm{G}_2$ structures on compact manifolds satisfying the known topological obstructions to holonomy $\mathrm{G}_2$ metrics

We construct new compact manifolds endowed with closed $\mathrm{G}_2$ structures that satisfy the topological properties found by Joyce and Baraglia for the existence of a torsion-free $\mathrm{G}_2$ structure in the same cohomology class. Those manifolds arise as resolutions of orbifolds of the form $M/\mathbb{Z}_2$, where $M$ itself does not admit any torsion-free $\mathrm{G}_2$ structure. We develop an equivariant resolution procedure for closed $\mathrm{G}_2$ orbifolds with $\mathbb{Z}_2$ isotropy, and analyze some topological properties of the resolved manifolds, including their first Pontryagin class and certain triple Massey products.