In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32.
Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element in $A$. If further $G/A$ is abelian, then $G$ has |G:A| linear characters, and $(|A|-1)/|G:A|$ non-linear irreducible characters of degree =|G:A| which vanish off $A$.
By the hypotheses, every conjugacy class contained in $A$ has order=|G:A|, except the trivial one. Moreover, we find that if $C$ is a class which contains one element in $\alpha A$, then $C$ is contained in $\alpha A$. Let $A$ act on $C$ by conjugation and partition $C$ into orbits. Again we find that no element in $C$ is fixed by $A$, so that |C| is greater than |A|, thus
k=the number of classes in $G$ is $\le 1+(|A|-1)/|G:A|+(|G|-|A|)/|A|$.
On the other hand, as $G' \subset A$, we find that the number of linear characters is $\ge |G:A|$. Furthermore, by Mackey's irreducibility criterion, there are exactly (|A|-1)/|G:A| irreducible characters induced by linear ones of $A$. Therefore, we conclude as stated.
As is obvious, this approach, if correct, exploits properties of induced characters of Mackey, with which I am still not so familiar, and hence I might ask:
I: Is my try valid?
II:How to proceed in an elementary manner?