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A concrete geometrisation scheme is proposed for the Green-Schwarz cocyclesin the supersymmetric de Rham cohomology of the super-minkowskian spacetime,which determine standard super-$\sigma$-model dynamics of super-p-branes. Thescheme yields higher-supergeometric structures with supersymmetry akin to thoseknown from the un-graded setting - distinguished (Murray-type) p-gerbe objectsin the category of Lie supergroups. These are shown to carry a canonicalequivariant structure for the action of the discrete Kosteleck\'y-Rabinsubgroup of the target supersymmetry group on the target, and thus to resolvethe `topology' of the corresponding Rabin-Crane soul superorbifold of thesuper-minkowskian spacetime. The equivariant structure is seen to effectivelydefine a novel $\sigma$-model of super-p-brane dynamics in the super-orbifoldthrough the construction of the corresponding (gauge-)symmetry defect.

Equivariant Cartan-Eilenberg supergerbes, the Kosteleck\'y-Rabin defect and descent to the Rabin-Crane superorbifold