TITLE: What is known about ordinary character values at involutions?
QUESTION [15 upvotes]: Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$. 
Let $x$ be an involution in $G$. 
I'd like to ask the following
Question 1:

What is known about $\chi(x)?$

1a) Are there criteria when $\chi(x)$ is positive / negative / zero ?
Of course, $1_{\text{Aut}(V)}=\rho(x^2)=\rho(x)^2$, such that the only possible eigenvalues of $\rho(x)$ are $\pm 1$. 
Moreover, there is an article written by P.X. Gallagher with the title "Character values at involutions" (DOI: https://doi.org/10.1090/S0002-9939-1994-1185260-1) dealing with the case that $\int_G\chi_1\chi_2\chi_3 \neq 0$, where the integral is in the sense of the Haar measure.
EDIT:
The main parts of Gallagher's results are the following ones:
For an involution $\sigma$ of a finite group $G$ and an irreducible complex representation $R$ of $G$, denote by $q$ the proportion of $-1$'s among the eigenvalues of $R(\sigma)$. Then:
$(*)$ $\frac{1}{h}\leq q \leq 1-\frac{1}{h}$, unless $q = 0$ or $1$,
where $h$ is the index of the centralizer $C$ of $\sigma$.
Moreover, if $\int_G\chi_1\chi_2\chi_3 \neq 0$, then $(*)$ is refined to prove that the proportions of $-1$'s among the eigenvalues of $\rho_1, \rho_2$ and $\rho_3$ (i.e., the corresponding representations) at $\sigma$ are the sides of a triangle on a sphere of circumference 2.
$\ $
1b) When does $\int_G\chi_1\chi_2\chi_3 \neq 0$ happen (necessary / sufficient criteria)?
1c) When does $\int_G\chi_1\chi_2\chi_3 = 0$ happen (necessary / sufficient criteria)?
1d) Are there results apart from Gallagher's result? 
1e) Can one deduce additional information, if all considered characters lie in the same 2-block?
Thanks for the help.

REPLY [3 votes]: Suppose that $\chi$ is an irreducible character of a finite group $G$ and $t$ is an involution in $G$. It is well known that $\chi(1)\equiv\chi(t)$ (mod $2$). Stephen Gagola and Sidney Garrison showed (J. Algebra 74 (1982) 20--51) that if $\chi$ is a faithful orthogonal character of $G$, and $\chi(1)-\chi(t)\equiv4$ (mod $8$), then $G$ has a non-trivial double cover. Moreover if $t$ lies in the commutator subgroup of $G$, then $H^2(G,{\mathbb C}^\times)$ has even order. They also have results related to the restriction of a real-valued character to a Klein-four subgroup of $G$. They used these results to verify that $M_{22}$ has a four-fold cover.