Detecting ends of residually finite groups in profinite completions

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Let be a variety of finite groups. We use profinite Bass--Serre theory to show that if u:H G is a map of finitely generated residually groups such that the induced map u:H→G is a surjection of the pro- completions, and G has more than one end, then H has the same number of ends as G. However if G has one end the number of ends of H may be larger; we observe cases where this occurs for 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:H→G 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.