  We introduce the notion of strong embeddability for a metric space. This
property lies between coarse embeddability and property A. A relative version
of strong embeddability is developed in terms of a family of set maps on the
metric space. When restricted to discrete groups, this yields relative coarse
embeddability. We verify that groups acting on a metric space which is strongly
embeddable has this relative strong embeddability, provided the stabilizer
subgroups do. As a corollary, strong embeddability is preserved under group
extensions.
