Detecting ends of residually finite groups in profinite completions

Let the class of finite p-groups.We produce a monomorphism of groups u : H ↪ G such that: either G is hyperbolic but not residually finite; or u : Ĥ → Ĝ is an isomorphism of profinite completions but H has property (T) (and hence (FA)), but G has neither. Either possibility would give new examples of pathological finitely generated groups.