  This article explores the interplay between the finite quotients of finitely
generated residually finite groups and the concept of amenability. We construct
a finitely generated, residually finite, amenable group $A$ and an uncountable
family of finitely generated, residually finite non-amenable groups all of
which are profinitely isomorphic to $A$. All of these groups are branch groups.
Moreover, picking up Grothendieck's problem, the group $A$ embeds in these
groups such that the inclusion induces an isomorphism of profinite completions.
  In addition, we review the concept of uniform amenability, a strengthening of
amenability introduced in the 70's, and we prove that uniform amenability
indeed is detectable from the profinite completion.
