  Given a homomorphism $\tau$ from a suitable finite group $\mathsf{\Gamma}$ to
$\mathsf{SU}(4)$ with image $\mathsf{\Gamma}^\tau$, we construct a
cohomological gauge theory on a noncommutative resolution of the quotient
singularity $\mathbb{C}^4/\mathsf{\Gamma}^\tau$ whose BRST fixed points are
$\mathsf{\Gamma}$-invariant tetrahedron instantons on a generally non-effective
orbifold. The partition function computes the expectation values of complex
codimension one defect operators in rank $r$ cohomological Donaldson-Thomas
theory on a flat gerbe over the quotient stack
$[\mathbb{C}^4/\,\mathsf{\Gamma}^\tau]$. We describe the generalized ADHM
parametrization of the tetrahedron instanton moduli space, and evaluate the
orbifold partition functions through virtual torus localization. If
$\mathsf{\Gamma}$ is an abelian group the partition function is expressed as a
combinatorial series over arrays of $\mathsf{\Gamma}$-coloured plane
partitions, while if $\mathsf{\Gamma}$ is non-abelian the partition function
localizes onto a sum over torus-invariant connected components of the moduli
space labelled by lower-dimensional partitions. When
$\mathsf{\Gamma}=\mathbb{Z}_n$ is a finite abelian subgroup of
$\mathsf{SL}(2,\mathbb{C})$, we exhibit the reduction of Donaldson-Thomas
theory on the toric Calabi-Yau four-orbifold
$\mathbb{C}^2/\,\mathsf{\Gamma}\times\mathbb{C}^2$ to the cohomological field
theory of tetrahedron instantons, from which we express the partition function
as a closed infinite product formula. We also use the crepant resolution
correpondence to derive a closed formula for the partition function on any
polyhedral singularity.
