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This is a continuation of a programme, initiated in Part I[arXiv:1706.05682], of geometrisation, compatible with the SUSY present, of theGreen-Schwarz $(p+2)$-cocycles coupling to the topological charges carried by$p$-branes on reductive homogeneous spaces of SUSY groups described byGS(-type) super-$\sigma$-models. In the present part, higher-geometricrealisations of the various SUSYs - both global and local - of these fieldtheories are studied at length in the form of - respectively - families ofgerbe isomorphisms indexed by the global-SUSY group and equivariant structureswith respect to SUSY actions amenable to gauging. The discussion, employing analgebroidal analysis of the small gauge anomaly, leads to a novel definition ofa supersymmetric equivariant structure on the Cartan-Eilenberg super-$p$-gerbeof Part I with respect to actions of distinguished normal subgroups of the SUSYgroup. This is exemplified by the ${\rm Ad}_\cdot$-equivariant structure on theGS super-$p$-gerbes for $\,p\in\{0,1\}\,$ over the super-Minkowski space, whoseexistence conforms with the classical results for the bosonic counterparts ofthe corresponding super-$\sigma$-models. The study also explores thefundamental gauge SUSY of the GS super-$\sigma$-model aka $\kappa$-symmetry.Its geometrisation calls for a transcription of the field theory to the dualtopological Hughes-Polchinski formulation. Natural conditions for thetranscription are identified and illustrated on the example of thesuper-Minkowskian model of Part I. In the dual formulation, the notion of anextended HP $p$-gerbe unifying the metric and topological degrees of freedom ofthe GS super-$\sigma$-model is advanced. Its compatibility with$\kappa$-symmetry is ensured by the existence of a linearised equivariantstructure. The results reported herein lend strong structural support to thegeometrisation scheme postulated in Part I.

Equivariant Cartan-Eilenberg supergerbes II. Equivariance in the super-Minkowskian setting