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Regarding a question which arose in comments to another answer: The lambda ring structure is on $RG$ is not enough to reconstruct the group. Dade has given examples (MathSciNet review here; paper does not appear to be available onlinehere) of pairs of groups which have the same character table with the same power maps, and from this it follows that the whole lambda ring structure is the same.

3 added 45 characters in body

Regarding a question which arose in comments to another answer: The lambda ring structure is on $RG$ is not enough to reconstruct the group. Dade has given examples (I do MathSciNet review here; paper does not have access appear to mathscinet here, but this is in the very first number of J. Alg.be available online) of pairs of groups which have the same character table with the same power maps, and from this it follows that the whole lambda ring structure is the same.

2 added 1 characters in body

Regarding a question which arose in comments to another answer: The lambda ring structure is on $RG$ is not enough to reconstruct the group. Dade has given examples (I do not have access to mathscinet here, but this is in the very first number of J. Alg.) of pairs of groups whichhave which have the same character table with the same power maps, and from this it follows that the whole lamnda lambda ring structure is the same.

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