Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$.

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.

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.