Let $G$ be a finite group and denote by $Z$ its centre. Let $H,K \leq G$ be two conjugate subgroups, i.e. there exists some $g \in G$ such that $K^g=H$. First of all, we get as an observation that $$K \cap Z=H^g \cap Z=H^g \cap Z^g=(H \cap Z)^g=H \cap Z,$$ since $H \cap Z$ is a central subgroup of $G$. Further $$K/K \cap Z\cong KZ/Z \leq G/Z,$$ so, if we assume that $G/Z$ is abelian, we get $$K/K \cap Z\cong KZ/Z=H^gZ/Z=(HZ/Z)^{gZ}=HZ/Z\cong H/ H \cap Z.$$ So in general $K/K \cap Z$ and $H/ H \cap Z$ will only be isomorphic, not equal. Assuming there is no error in this argument, I want to ask if there is some set of extra conditions we can impose on $G$ so that whenever two subgroups $H,K$ of $G$ are conjugate, we have equality between $K/K \cap Z$ and $H/ H \cap Z$.
Many thanks.
