The regularity of quotient paratopological groups

Let $H$ be a closed subgroup of a regular abelian paratopological group $G$. The group reflexion $G^{\mathrm{♭}}$ of $G$ is the group $G$ endowed with the strongest group topology, weaker that the original topology of $G$. We show that the quotient $G\mathrm{/}H$ is Hausdorff (and regular) if $H$ is closed (and locally compact) in $G^{\mathrm{♭}}$. On the other hand, we construct an example of a regular abelian paratopological group $G$ containing a closed discrete subgroup $H$ such that the quotient $G\mathrm{/}H$ is Hausdorff but not regular.