Result: Counterexamples to a conjecture about the sum of degrees of irreducible characters

Let \(\text{Irr}(G)\) be the set of all irreducible complex characters of a finite group \(G\). If \(k(G)\) is the class number of \(G\), then \(| \text{Irr} (G) | = k(G)\). Let \(T(G) = \sum_{\chi \in \text{Irr}(G)} \chi(1)\), \(f(G) = | G|^{-1} \times T(G)\). If \(H \leq G\) then \(f(H) \geq f(G)\) (this is an easy corollary of Frobenius reciprocity). Under some suppositions on degrees of irreducible characters the following inequality holds [see \textit{K. G. Nekrasov} and the author, ibid. 33, 333-354 (1986; Zbl 0649.20005)]: \(f(G/H) \geq f(G)\) for all normal subgroups \(H\) in \(G\) [see loc. cit.; Lemma 2.6]. The following conjecture was posed [see question 4.3 in ibid.]: (*) If \(H\) is normal in \(G\) then \(f(G/H) \geq f(G)\). In this note we give counterexamples to (*).