Question about Suzuki's theory of exceptional characters

Question

$\DeclareMathOperator\Irr{Irr}$As elegant as Suzuki's theory is, the set up requires that the number of conjugacy classes of $p$-elements in a cyclic T.I. (as an example) Sylow $p$-subgroup $P$ of $G$, $t$, is at least 2, in order to produce at least two exceptional characters of $N_G(P)$ of equal degrees which in turn generates a virtual character which vanishes off $p$-elements.

My question is, what happens when $t = 1$? Is there an analogous theory to treat this case since the end result is that $\Irr(G)$ is still largely controlled by $N_G(P)$? (For instance McKay's conjecture still holds.)

To illustrate, the theory of exceptional characters would work perfectly for $G = A_5$ for $p = 5$ since $N_G(P) \cong D_{10}$ have two exceptional characters of degree 2. However when $G = S_5$, it seems that the theory won't work since in this case $N_G(P) \cong C_5 \rtimes C_4$ has only 1 non-linear character, of degree 4. But $\Irr(G)$ and $\Irr(N_G(P))$ are strongly connected in this case. Is there a similar theory that addresses cases like $G = S_5$ with $p = 5$? I'd be grateful to know any literature on such theory if it exists.

And I suppose when $t=1$ it's not always possible to find a normal subgroup $N$ of $G$ for which the theory of exceptional characters would work, as in the $A_5 \unlhd S_5$ case. Is that right?

(I just learned this website is for research-level questions, so I moved the question from Math Stack Exchange.)

Answer 1

The theory of blocks with cyclic defect group addresses this question in slightly more generality. The beginning of this theory was Richard Brauer's Annals paper in the early 1940s which determined the structure of principal blocks of defect $1$, and showed, in particular that there is a bijection between irreducible characters in the principal $p$-block of $G$ and the principal $p$-block of $N_{G}(P)$, where $P$ is a Sylow $p$-subgroup of $G$ (of order $p$). The full character theory of $p$ blocks with cyclic defect groups was completed circa 1967 by E.C. Dade, following earlier advances by J.A. Green and J.G. Thompson. However, the full character-theoretic conclusions require methods from modular representation theory (at present), and (as far as I am aware), can't be deduced purely by ordinary (i.e., complex) character-theoretic methods. Expositions of the cyclic defect theory can be found in many modern texts on representation theory (perhaps texts by M. Isaacs and G. Navarro (and maybe also one by D. Goldschmidt) use mainly complex character-theoretic methods in their treatment of the cyclic defect theory).