Following on from some of myprevious MO questions on finite group theory... $\newcommand{\Irr}{\operatorname{Irr}}\newcommand{\Conj}{\operatorname{Conj}}\newcommand{\AMZL}{{\rm AM}_{\rm Z}}$

Let $G$ be a finite group, $\Irr$ the set of irreducible characters (working over $\mathbb C$) and $\Conj$ the set of conjugacy classes. Consider the following quantity, which seems to have first been introduced in work of Azimifard, Samei and Spronk $$\alpha_G = \frac{1}{|G|^2} \sum_{\phi\in\Irr} \sum_{C\in\Conj} \phi(e)^2 |\phi(C)|^2 |C|^2 . $$ It is not hard to show, just using basic character theory, that $\alpha_G \geq 1$ with equality if and only if $G$ is abelian. This constant/invariant arises as a minorant for a more interesting invariant of $G$ which was studied in the aforementioned paper.

Call this "more interesting invariant" $\AMZL(G)$. It turns out that $\AMZL(G)\geq \AMZL(G/N)$ for every normal subgroup $N$, a property which is vital to some work I am writing up that obtains lower bounds on $\AMZL(G)$. One branch of the case-by-case attack I'm using works by getting lower bounds on $\alpha_G$ when $G$ has trivial centre, and so if $\alpha_G$ never increases when we replace $G$ with a quotient then I could in fact get general lower bounds on $\alpha_G$ by inductively modding out by the centre.

Question.Do there exists a finite group G and a normal subgroup N for which $\alpha_{(G/N)} > \alpha_G$ ?

I did try for a while to show this can never happen but rapidly got stuck, so I'm hoping that MO readers who are more skilled/experienced with finite groups can either suggest counterexamples to try, or give some evidence that no such counterexamples exist. Please bear in mind that I have next to no experience with GAP or similar, so I don't mind concrete suggestions of code, but responses of the form "get GAP to look through all non-simple non-abelian examples of order < 100" may not be as helpful as their authors imagine.

Just in case it helps to rule out certain avenues of attack: if $G$ is a group with two character degrees (i.e. there is an integer $m$ such that $\phi(e)\in \{1,m\}$ for all $\phi\in\Irr$) then one can evaluate $\alpha_G$ explicitly, using ideas similar to this paper: one obtains

$$ \alpha_G - 1 = (m^2-1) \left( 1- \frac{|L|}{|G|^2} \sum_{C\in\Conj} |C|^2 \right) $$ where $|L|$ is the number of linear characters on $G$.

In particular, suppose if $G=H \rtimes C_2$ is a generalized dihedral group with $|H|=2n+1$ ($n$ a positive integer). Back of the envelope scribbling gives me $m=2$, $|L|=2$ and the conjugacy classes have sizes $1$, $2$ repeated $n$ times, and $2n+1$, so that $$ \frac{\alpha_G -1}{3} = 1 - \frac{1}{2(2n+1)^2}(1+ 4n + (2n+1)^2) = \frac{1}{2}\left(1-\frac{1}{2n+1}\right)^2 $$ It also seems to me that proper quotients of $G$ must either be abelian or generalized dihedral with smaller order, in which case we would have $\alpha_G \geq \alpha_{(G/N)}$ for every $N \lhd G$.

iff$G$ is abelian, it's enough to use any $G,N$ where $G/N$ is abelian but $G$ is not. For example, $G=S_n$ and $N=A_n$ for any $n\geq 3$, or $G$ the quaternion group and $N$ its center. $\endgroup$