Let $G$ be a finite group. $N$ a normal subgroup of $G$ and $K$ a characteristic subgroup of $N$: $$K \text{ char } N \triangleleft G.$$ If $Z(G/N)=1$ and $Z(G)=1$, does it follow that $Z(G/K)=1$?

As it is not entirely clear where the containments are meant to be, I will assume you are asking about the case $N\triangleleft K\triangleleft G$; in this case $Z(G) = Z(G/N) = 1$ is not sufficient in general to guarantee $Z(G/K)$ is also trivial. As an example, take $N$ to be the Klein4 subgroup of $S_4$ and $K = A_4$. Then $G/N\cong S_3$ so $Z(G) = Z(G/N)$ is trivial, but $G/K$ is abelian (isomorphic to $\mathbb{Z}/2\mathbb{Z}$) and so $Z(G/K)$ is not trivial. 

