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 [here][2]) 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
  [2]: http://dx.doi.org/10.1016/0021-8693(64)90002-X