Groupoids and relations among Reidemeister and among Nielsen numbers

New results include a new formula generalizing of the averaging formula in both the algebra and the geometry. We also generalize the original coincidence averaging formula for oriented infra-nilmanifolds to the smooth non-orientable category, and also to a pair of self-maps of smooth infra-solvmanifolds of type R. Other generalizations concern the finiteness of Reidemeister numbers. Finally we fill in proofs and details missing from previous work.