Explicit construction of induced character from cyclic subgroup of symmetric group

Question

I have been computing some characters by hand, but just can't seem to figure out how they relate to the standard "induced character" constructions I can find.

Small example: For $G=S_4$ and the cyclic subgroup $H=\langle(1234)\rangle$, references indicate that the permutation character (well, character of the permutation representation) on the coset space $G/H$ should be obtained by inducing the trivial character $\chi$ on $H$, which of course has value one for every cycle class.

I can compute this permutation character (call it $\psi$) by actually counting fixed points, of course, and as far as I can tell it has value $\psi(e)=6$ (as expected!!!), $\psi(C_2^2)=2$, $\psi(C_4)=2$, and zero elsewhere.

But using the formula $$\psi(g)=\chi^{S_4}_H(g)=\frac{1}{4}\sum_{x\in S_4\\ xgx^{-1}\in H} 1$$ (e.g. Bump's notes, other sources just use coset reps and don't average over $|H|$) I get nowhere, because clearly no element of the form $C_2^2$ is conjugate with a $4$-cycle – the cool thing about the symmetric group is conjugacy classes are precisely cycle types.

Clearly I still either have a serious misunderstanding of how induced characters/representations work, or I have made some really obvious computation error in my fixed points (or both). I'd be grateful for any correction, and in particular for explicit computation of whatever I am messing up here.

Answer 1

$H$ contains the element $(13)(24)$ (the square of the generator $(1234)$), which is of type $C_2^2$.