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.