On subgroups of the form $HZ(G)$ where $H$ is abelian subgroup of non-abelian group $G$ such that $H \rlap{\;\,/}\subseteq Z(G)$

Question

Let $H$ be a an abelian subgroup of a non-abelian group $G$ such that $H \rlap{\;\,/}\subseteq Z(G) Z(G)$ ; then I can prove that $HZ(G)$ is an abelian subgroup such that $Z(G) \subset HZ(G) \subset G$ ; also if $H$ is normal then so is $HZ(G)$ as $gHZ(G)g^{-1} \subseteq gHg^{-1} Z(G) \subseteq HZ(G)$ ; I would like to know when are two subgroups $H_1Z(G) $ ; $H_2Z(G)$ isomorphic ? or more generally when are $H_1Z(G_1) $ ; $H_2Z(G_2)$ isomorphic given $G_1,G_2$ is isomorphic ( here $H_1$ a subgroup of $G_1$ , $H_2$ a subgroup of $G_2$ ) ?