Let $G$ be a finite group. In an an earlier question, Fedor asked whether the square root counting function $r_2:G\rightarrow \mathbb{N}$, which assigns to $g\in G$ the number of elements of $G$ that square to $g$, attains its maximum at the identity element, when $G=S_n$. I gave an affirmative answer using representation theory, which is valid when $S_n$ is replaced by any group that has no symplectic representations. In other words, the argument I gave works whenever any irreducible complex representation of $G$ is either realisable over $\mathbb{R}$ or has non-real valued character. Again in other words (and closest to the actual argument) I need the Frobenius-Schur indicators of all irreducible complex representations to be non-negative.

This is the first of two follow-up questions (the second one being about $n$-th roots), which Pete Clark encouraged me to ask here. I ran a quick computer experiment. There are 1911 groups among the groups of size up to 150 that have a symplectic representation. In 1675 of them, the square root counting function does not attain its maximum at the identity element. Is there a nice (representation theoretic?) criterion that singles out the 300-odd remaining groups? A criterion that includes the previous one as a special case would of course be particularly interesting. in other words, I am asking:

what does it tell you about the group (about its representation theory) if the square root counting function attains its maximum at the identity?

Any brain storming ideas or heuristics are welcome. Criteria that catch some of the remaining groups, even if not all, are also of interest. Thank you in advance!

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