  We investigate the validity of the isometry extension property for
(Riemannian) Einstein metrics on manifolds with boundary. Given a metric on the
boundary, this is the issue of whether any Killing field of the boundary metric
extends to a Killing field of any bulk or filling Einstein metric inducing the
given data on the boundary. Under a mild condition on the fundamental group,
this is proved to be the case at least when the Killing field preserves the
mean curvature of the boundary.
