Let ${\rm Irr}(G)$ be the set of complex irreducible characters of a finite group $G$. The Frobenius-Schur indicator of $\chi\in{\rm Irr}(G)$ is defined to be $\epsilon(\chi):=\frac{1}{|G|}\sum_{g\in G}\chi(g^2)$. It is known that $\epsilon(\chi)=-1,0,1$ and that $\epsilon(\chi)\ne0$ if and only if $\chi$ is real valued. Moreover $$ \sum\limits_{\chi\in{\rm Irr}(G)}\hspace{-.1cm}\epsilon(\chi)\chi(g)=|\{x\in G\mid x^2=g\}|,\quad\mbox{for all $g\in G$.} $$
Consider the following class functions on $G$: $$ a(g):=\sum_{\chi\in{\rm Irr}(G)}\frac{\chi(g)}{\chi(1)},\quad b(g):=\sum_{\chi\in{\rm Irr}(G)}\frac{\epsilon(\chi)\chi(g)}{\chi(1)},\quad c(g):=\sum_{\chi\in{\rm Irr}(G)}\frac{\epsilon(\chi)^2\chi(g)}{\chi(1)}. $$ For obvious reasons the average value of each of $a,b,c$ is $1$. Moreover it is known that $$ \begin{aligned} |G|a(g)&=|\{x,y\in G\mid x^{-1}y^{-1}xy=g\}|\quad\mbox{and}\\ |G|c(g)&=|\{x,y\in G\mid xy^{-1}xy=g\}|. \end{aligned} $$ So $|G|a$ and $|G|c$ take non-negative integer values. In the few cases I checked, so too does $|G|b$.
Question: does $|G|b$ count anything? More specifically, is there a 2-variable word $w$ such that $|G|b(g)=|\{x,y\in G\mid w(x,y)=g\}|$ for all $g\in G$?
