This question occured to me when I was working with upper central series:
Let $K\le N$ be normal subgroups of $G$. Then there is some normal subgroup $M$ of $G$ satisfying $M/N = Z(G/N)$. I claim that
$$Z\left( \frac{G/K}{N/K} \right) = \frac{M/K}{N/K}$$
I wrote down all the details of my claim using cosets. But that is too tedious.
I am looking for alternative ways of proving this claim. Is there any result I can use to make it follow easily?