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][1]; 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.


  [1]: http://www.ams.org/mathscinet-getitem?mr=170957