  A complete and natural geometric and physical interpretation of the
tangential gauge supersymmetry, also known as $\kappa$-symmetry, of a large
class of Green-Schwarz(-type) super-$\sigma$-models for the super-$p$-brane in
a homogeneous space of a (supersymmetry) Lie supergroup is established in the
convenient setting of the topological Hughes-Polchinski formulation of the
super-$\sigma$-model and illustrated on a number of physical examples. The
supersymmetry is identified as an odd superdistribution in the tangent sheaf of
the supertarget of the super-$\sigma$-model, generating - through its weak
derived flag - the vacuum foliation of the supertarget. It is also demonstrated
to canonically lift to the vacuum restriction of the extended Hughes-Polchinski
$p$-gerbe associated with the superbackground of the field theory, and that in
the form of a canonical linearised equivariant structure thereon, canonically
compatible with the residual global supersymmetry of the vacuum.
