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Thank you for the answer. I don't quite understand the last line. WHat about cohomology class corresponding to $e\times Y$ itself? Then isn't the "G-invariantization" |G|-times the original one?
@Chris π∗ should be the transfer map. If my memory serves, one of the equalities in the last line holds. It seems that Russel's example below shows us the latter does not hold in general...
