Suppose $P\in Syl_p(G)$ is self-centralizing of order $p$. Let $[N_G(P):P]=e$. Then $G$ has at most $e+(p-1)/e$ irred. characters with $p'$-degree. This is Problem 7.6 in M Isaacs' Character Theory of Finite Groups. I'm given $P=\langle x\rangle\in Syl_p(G)$ of order $p$ such that $P=C_G(P)$. Let $N=N_G(P)$ and suppose $[N:P]=e$, which clearly divides $p-1$. Our objective is to show that $G$ has at most $e+(p-1)/e$ irred. characters of degree coprime to $p$.
Any subgroup of $G$ which is a T.I. set (trivial intersection), proper in its own normalizer, and contains the centralizer of every non-identity element is called a T.I.F.N. (trivial intersection Frobenius normalizer) subgroup of $G$. $P$ is T.I.F.N. if $e>1$. If further $1<e<p-1$, Corollary 7.18. of Brauer-Suzuki implies that $\mathscr{S}=\{\chi\in\mathrm{Irr}(N):P\nleq\mathrm{ker}(\chi)\}$ (where $N$ is a Frobenius group with Frobenius kernel $P$.) is coherent (with resepect to induction to $G$). So, the induction map defined on the degree-zero subspace of $\mathbb{Z}[\mathscr{S}]$ can be extended to a linear isometry $*$ on $\mathbb{Z}[\mathscr{S}]$. Moreover, $\chi(1)=e$ for all $\chi\in\mathscr{S}$ and $|\mathscr{S}|=(p-1)/e\geq2$. (It is rather trivial to construct the character table of $N$.)
Fix a $\chi_0\in\mathscr{S}$. Now, if $\psi\in \mathrm{Irr}(G)-\mathscr{E}$, where $\mathscr{E}$ denotes the set of exceptional characters associated with $\mathscr{S}$ and $*$. Lemma 7.19(a) shows that $$\psi_N=[\psi_N,\chi_0]\sum_{\chi\in\mathscr{S}}\chi+\vartheta$$ where $[\vartheta,\chi]=0$ for all $\chi\in\mathscr{S}$.
Clearly $\psi(1)=[\psi_N,\chi_0](p-1)+\vartheta(1)$, whereas$$\psi(x)=[\psi_N,\chi_0]\sum_{\chi\in\mathscr{S}}\chi(x)+\vartheta(x)=-[\psi_N,\chi_0]+\vartheta(x).$$Thus, $$p\mid(\psi(1)-\psi(x))=p[\psi_N,\chi_0]$$ as $\vartheta(x)=\vartheta(1)$.
The author hinted that one should show that $\sum_{\psi\in\mathscr{E}}|\psi(x)|^2\geq p-e$ (equivalently $\sum_{\psi\in\mathrm{Irr}(G)-\mathscr{E}}|\psi(x)|^2\leq e$). I think I know $\sum_{\psi\in\mathscr{E}}|\psi(x)|^2$ is $\mathscr{G}$-invariant, where $\mathscr{G}=Gal(\mathbb{Q}(\zeta_{|G|})/\mathbb{Q})$, & therefore is a rational integer. Since $0\leq\sum_{\psi\in\mathscr{E}}|\psi(x)|^2<p$, it remains to compute $\sum_{\psi\in\mathscr{E}}\psi(1)^2\pmod p$. (Note that I never claimed that $p\mid(\psi(1)-\psi(x))$ for $\psi\in\mathscr{E}$.) Why is it that the least residue of $\sum_{\psi\in\mathscr{E}}\psi(1)^2$ modulo $p$ should be $\geq p-e$? I'm absolutely clueless.
(Edit 1) Just realized that coherence implies Theorem 7.20(d), so that $\psi(1)$ is independent of the choice of $\psi\in\mathscr{E}$. By Thm. 7.20(a), this $\psi(1)$ is the degree of an irred. character of $G$ which is non-constant on $P-\{1\}$. Not sure what to say about this.
 A: Thm. 2.13 in Chap. 5 of M. Suzuki's Group Theory tells us that$$\sum_{g\in P-\{1\}}|\psi(g)|^2\geq e(p-e)$$ for all $\psi\in\mathscr{E}$. Summing up over all of $\mathscr{E}$ and noting that$$\sum_{\psi\in\mathscr{E}}|\psi(g)|^2=p-\sum_{\psi\in\mathrm{Irr}(g)-\mathscr{E}}|\psi(g)|^2=p-\sum_{\psi\in\mathrm{Irr}(g)-\mathscr{E}}|\psi(g^\sigma)|^2=\sum_{\psi\in\mathscr{E}}|\psi(g^\sigma)|^2$$for all automorphisms $\sigma$ of $P$, (second equality is due to the fact that non-exceptional characters are constant on $P-\{1\}$.) we see that $\sum_{\psi\in\mathscr{E}}|\psi(x)|^2\geq p-e$ or equivalently $\sum_{\psi\in\mathrm{Irr}(G)-\mathscr{E}}|\psi(x)|^2\leq e$. This & the fact that $p\mid (\psi(1)-\psi(x))$ for all nonexceptional $\psi$ shows that there are at most $e$ nonexceptional irred. characters with $p'$-degree. The result follows.
The case $e=p-1$ is similar since here $p\mid (\psi(1)-\psi(x))$ for all $\psi\in \mathrm{Irr}(G)$ by noting that $p\mid (\chi(1)-\chi(x))$ for all $\chi\in \mathrm{Irr}(N)$. In the case $e=1$, $G$ has a normal $p$-complement(, which is a Frobenius kernel of $G$) by transfer. Here, all the nonlinear characters of $G$ are induced from the Frobenius kernel.
(Edit 1) For the sake of completeness, I worked out the proof of Suzuki's result: Let $\pm\chi^*\in\mathscr{E}$. Then $\chi^*-\chi$ is an integral constant $m$ on $P-\{1\}$ by Thm. 7.20(b). So,\begin{align}
\sum_{g\in P-\{1\}}|\chi^*(g)|^2 & =\sum_{g\in P-\{1\}}|\chi(g)+m|^2\\      & =\sum_{g\in P-\{1\}}|\chi(g)|^2+m^2+2m\mathrm{Re}\chi(g) \\ 
& =\sum_{g\in P}|\chi(g)|^2 -\chi(1)^2+(p-1)m^2+2m\mathrm{Re}(\sum_{g\in P}\chi(g)-\chi(1))\\& =ep-e^2+(p-1)m^2-2me\geq e(p-e),
\end{align}
where the cases $m\geq0$ and $m<0$ are treated separately.
