Quotient group of finite CN-group by solvable normal subgroup is CN-group

Statement

Suppose $G$ is a finite CN-group (i.e., the centralizer of every non-identity element in $G$ is nilpotent). Suppose $H$ is a solvable normal subgroup of $G$. Then, the quotient group $G/H$ is a CN-group as well.

References

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 400, Lemma 14.1.3, ^{More info}