  We prove a broad generalization of a theorem of W. Burnside on real
characters using permutation characters. Under a necessary hypothesis, We can
give some control on multiplicities (a result that needs the Classification of
Finite Simple Groups). Along the way, we also give a new characterization of
the 2-closed finite groups using odd-order real elements of the group. All this
can be seen as a contribution to Brauer's Problem 11. We also obtain similar
results for 2-Brauer characters. We also classify finite primitive permutation
groups in which every real element has a fixed point.
