Elements Inverted by an Automorphism and the Sum of the Character Degrees (Kourovka Notebook Problem 16.60)

For a finite group \(G\) let \(T(G)\) be the sum of the degrees of the irreducible complex characters of \(G\), and for an automorphism \(\alpha\) of \(G\) let \(S_\alpha=\{g\in G:\alpha(g)=g^{-1}\}\). D. MacHale asked in the Kourovka Notebook (Problem 16.60) whether \(|S_\alpha|\le T(G)\) for every \(\alpha\in\operatorname{Aut} G\). The inequality is stated as Theorem 3.1 of a 2026 preprint of Yerrapati, Dixit and Shukla; the proof given there for the bound on twisted indicators that it uses (Lemma 2.19 there) is incomplete. We give a complete proof, in two short ways. The first follows the strategy of that preprint: \(|S_\alpha|=\sum_\chi\iota_\alpha(\chi)\chi(1)\) with \(\iota_\alpha(\chi)=|G|^{-1}\sum_{g\in G}\chi(g\alpha(g))\), and we prove \(|\iota_\alpha(\chi)|\le1\) for every automorphism from the orthogonality relations for matrix coefficients. The second restricts \(\alpha\) to the subgroup of fixed points of \(\alpha^2\), on which it induces an automorphism of order at most \(2\), uses the twisted Frobenius–Schur count of Kawanaka and Matsuyama for involutory automorphisms, and then the inequality \(T(H)\le T(G)\) for subgroups \(H\). We also show that equality holds if and only if \(\alpha^2=1\) and all twisted indicators are equal to \(1\); in particular the inequality is strict when \(\alpha\) has order at least \(3\). Special cases were known: involutory automorphisms (from the twisted Frobenius–Schur count), inner automorphisms and complete groups (Bors, 2019), and abelian groups, dihedral groups and groups of order \(p\), \(2p\), \(p^2\) (Yerrapati, Dixit and Shukla, 2026). The novelty is modest: a complete proof of the general case and the characterisation of equality. This is an unrefereed note.