  Recent work applying higher gauge theory to the superstring has indicated the
presence of `higher symmetry'. Infinitesimally, this is realized by a `Lie
2-superalgebra' extending the Poincare superalgebra in precisely the dimensions
where the classical supersymmetric string makes sense: 3, 4, 6 and 10. In the
previous paper in this series, we constructed this Lie 2-superalgebra using the
normed division algebras. In this paper, we use an elegant geometric technique
to integrate this Lie 2-superalgebra to a `Lie 2-supergroup' extending the
Poincare supergroup in the same dimensions.
  Briefly, a `Lie 2-superalgebra' is a two-term chain complex with a bracket
like a Lie superalgebra, but satisfying the Jacobi identity only up to chain
homotopy. Simple examples of Lie 2-superalgebras arise from 3-cocycles on Lie
superalgebras, and it is in this way that we constructed the Lie 2-superalgebra
above. Because this 3-cocycle is supported on a nilpotent subalgebra, our
geometric technique applies, and we obtain a Lie 2-supergroup integrating the
Lie 2-superalgebra in the guise of a smooth 3-cocycle on the Poincare
supergroup.
