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.