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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 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.

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 here) 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.

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 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.

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 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.

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 the same character table with the same power maps, and from this it follows that the whole lamnda ring structure is the same.

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 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.