We introduce a relative fixed point property for subgroups of a locally
compact group, which we call relative amenability. It is a priori
weaker than amenability. We establish equivalent conditions, related
among others to a problem studied by Reiter in 1968. We record a solution to Reiter's problem.
We then study the class X of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that X contains all familiar groups. Actually, no group is known to lie outside X.
Since relative amenability is closed under Chabauty limits, it follows that any Chabauty limit of amenable subgroups remains amenable if the ambient group belongs to the vast class X.