This is the second follow-up to this question on square roots of elements in symmetric groups and is concerned with generalisations to $n$-th roots. Let $G$ be a finite group and let $r_n(g)$ be the number of elements $h\in G$ such that $h^n = g$. In other words, $$r_n(g) = \sum_{h\in G}\delta_{h^n,g},$$ where $\delta$ is the usual Kronecker delta. In a comment to my answer to the above mentioned question, Richard Stanley notes that if $G=S_m$, then $r_n(g)$ attains its maximum at the identity element of $G$. My question is: how far does this generalise and what exactly does it tell us about $G$? This should be primarily a question about higher Frobenius-Schur indicators. Let me elaborate a bit.

The function $r_n$ is clearly a class function on $G$ and, upon taking its inner product with all irreducible characters of $G$, one finds that $$r_n(g) = \sum_\chi s_n(\chi)\chi(g),$$ where the sum runs over all irreducible complex characters of $G$ and $s_n(\chi)$ is the $n$-th Frobenius-Schur indicator of $\chi$, defined as $$s_n(\chi) = \frac{1}{|G|}\sum_{h\in G}\chi(h^n).$$ When $n=2$, the Frobenius-Schur indicator is equal to 0,1 or -1 and carries explicit information about the field of definition of the representation associated with $\chi$.

What do higher Frobenius-Schur indicators tell us about the representations and, by extension, about the group? What do we know about their values? Have higher Frobenius-Schur indicators been studied in any detail?

For additional focus:

Given $n\in \mathbb{N}$, for what groups $G$ do we have $\max_g \; r_n(g) = r_n(1)$? For what groups does this hold for

all$n$?

As noted by Richard Stanley, the latter is true for all symmetric groups. It is also easy to see that the set of groups with this property is closed under direct products, and that all finite abelian groups possess this property.