OK, I've given you a reduced road map given your new stipulations.

I see your problem now. I hadn't realised exactly what you were asking for. I still think it shouldn't be too difficult. It should follow from the fact that $-1 = w_0\sigma$ but I'll have to think about it. Sorry it's not yet a complete answer.

Thanks. That's a good point! I really like Isaacs' book but it can be pretty challenging for someone with only a passing interest in character theory.

I would suggest this is very much related to the fact that the longest element of the Weyl group of type E6 is $-\sigma$ where $\sigma$ is the automorphism of the root system induced by the unique non-trivial automorphism of the Dynkin diagram. Thus the longest element acts as $-1$ on anything fixed by the diagram automorphism. The fixed point subgroup under $\sigma$ is the group obtained by the diagram fold, which is just the Weyl group of type F4.

Just curious but is this not the main aim of Rob Wilson's book "The finite simple groups"? On page 3 he states "It is the chief aim of this book to explain, as far as space allows, the statement of CFSG. Thus we seek to introduce all the finite simple groups, to provide concrete constructions whenever possible, to calculate the orders of the groups, prove simplicity, ..."