Let $G$ and $H$ be two groups such that $f:G/Z(G)\rightarrow H/Z(H)$ be an isomorphism. Now let $A$ and $B$ be normal subgroups of $G$ and $H$ respectively. If I define a map by $g:\frac{G}{A}\rightarrow\frac{H}{B}$ by $g(xA)=f(x)B$ then what conditions on $g$ should be imposed so that it becomes an isomorphism, where $f(x)$ is the image of $xZ(G)$ under $f$ without consedring the quotient.
My attempt The map $g$ should be well defined, for this $A\subseteq Z(G)$, $B\subseteq Z(H)$ and $f(AZ(G))\subseteq BZ(H)$ should be the necessary conditions. If the map is well defined then there is no problem in being an homomorphism. Now please give me some idea to find minimal condition on $f$ and $g$ so that $g$ become bijective.