  We develop the theory of simplicial extensions for bundle gerbes and their
characteristic classes with a view towards studying descent problems and
equivariance for bundle gerbes. Equivariant bundle gerbes are important in the
study of orbifold sigma models. We consider in detail two examples: the basic
bundle gerbe on a unitary group and a string structure for a principal bundle.
We show that the basic bundle gerbe is equivariant for the conjugation action
and calculate its characteristic class; we show also that a string structure
gives rise to a bundle gerbe which is equivariant for a natural action of the
String 2-group.
