Between the center of a quotient group and the total center i have a question on centers of a group and quotients.
Consider $G$ a group and $H$ any normal subgroup of $G$. Is there any relation between the center of the quotient $Z\left(G/H\right)$ and the total center $Z(G)$? I am looking for some kind of isomorphism between $Z\left(G/H\right)$ and a quotient on $G$ involving $Z(G)$ but i cannot figure it out. Maybe there is just an injection?
Thanks in advance!

What is sure is that you have $Z(G)/Z(G)\cap H$ is isomorphic to a subgroup of $Z(G/H)$ (see Nicky Hekster's answer for details about this). Apart from this, you cannot say anything in general.

$Z(G/H)$ is trivial whereas $Z(G)$ is not.

Consider a direct product $G=A \times S$ where $S$ is a non-abelian simple group and $A$ is an abelian group. It is not hard to see that $Z(G)=A\times\{1_S\}$. Furthermore, taking $H:=Z(G)$ then $G/H$ is naturally isomorphic to $S$ which has trivial center.

$Z(G/H)$ is equal to $G/H$ whereas $Z(G)$ is trivial.

Consider $S_3$ the symmetric group over $3$ elements. Then $Z(S_3)$ is trivial. Consider $S_3/A_3$ (where $A_3$ is the alternating group) then $S_3/A_3$ is isomorphic to $\mathbb{Z}/2$ and thus is abelian.
