I think that there is a mispriprint in the formulation of problem $1.D.19$ from Isaacs's book Finite Group Theory (FGT). I give it here
Let $F = F(G)$, where $G$ is an arbitrary finite group, and let $C = C_G(F)$. Show that $G/(G \cap F)$ has no non-trivial abelian normal subgroup.
Here $F(G)$ is a Fitting subgroup of $G$. I think that there should be $G/(F\cap C)$ instead of $G/(G\cap F)$ since $G \cap F = F$ and in this case the result doesn't depend on $C$. On the other hand if the correct formulation is $G/(F \cap C)$, then the result is not valid for example for $G =D_8$. Since in this case $G=F$, $C = Z(G)$ and $G/(F\cap C)$ is isomorpic to $C_2 \times C_2$, wich of course has nontrivial abelian normal subgroup.
So, what is the correct formulation of this problem?
As it is mentioned in the comments the correct formulation is $C/(C \cap F)$ has no nontrivial abelian normal subgroup. To prove this you may use the following hints:
Let $G$ be solvable then $G$ is non trivial if and only if $\mathbf{F}(G)$ is non trivial.
If $G$ is a finite group then $\mathbf{F}(G/\mathbf{Z}(G))= \mathbf{F}(G)/\mathbf{Z}(G).$
Show that $\mathbf{Z}(\mathbf{C}_G(\mathbf{F}(G)))= \mathbf{F}(\mathbf{C}_G(\mathbf{F}(G)))$ and apply Hint 2 in $\mathbf{C}_G(\mathbf{F}(G))$ and conclude that $\mathbf{F}(C/C \cap F)$ is trivial.