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Ok, I'm new to this so i'm sorry if this is abit warbled!...

We have a normal subgroup N of a finite group H such that [H:N]=2

We have a chatacter Chi belonging to the irreducible characters of H, Irr(H) , which is zero on H\N.

I have already shown that Chi restricted to N = Theta 1 + Theta 2, Where

Theta 1 is not equal to Theta 2, and both Theta 1, and Theta 2 belong to Irr(N).

I am now trying to show that given a conjugacy class K of H which is also a conjugacy class of N, we get Theta 1 (n) = Theta 2 (n) for n a member of K.

Intuitively, I can see this is true, I have seen examples in the alternating group A6, the normal subgroup of index 2 in S6 fro example.

I would be really greatful for a push in the right direction...I have books: 'James and Liebeck' and 'Isaacs'... if you could even direct me to some relevant info?

Wow, think my notation is abit crap, sorry.