Extending group actions on metric spaces

We address the following natural extension problem for group actions: Given a group [Formula: see text], a subgroup [Formula: see text], and an action of [Formula: see text] on a metric space, when is it possible to extend it to an action of the whole group [Formula: see text] on a (possibly different) metric space? When does such an extension preserve interesting properties of the original action of [Formula: see text]? We begin by formalizing this problem and present a construction of an induced action which behaves well when [Formula: see text] is hyperbolically embedded in [Formula: see text]. Moreover, we show that induced actions can be used to characterize hyperbolically embedded subgroups. We also obtain some results for elementary amenable groups. </jats:p>