This survey of profinite group completions shows which group properties are determined by the set of finite quotients, collecting both negative examples and positive results.
Baumslag,Residually finite groups with the same finite images, Compositio Math.29(1974), 249–252
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Infinite groups from the profinite point of view
This survey of profinite group completions shows which group properties are determined by the set of finite quotients, collecting both negative examples and positive results.