  It was observed recently that for a fixed finite group $G$, the set of all
Drinfeld centres of $G$ twisted by 3-cocycles form a group, the so-called group
of modular extensions (of the representation category of $G$), which is
isomorphic to the third cohomology group of $G$. We show that for an abelian
$G$, pointed twisted Drinfeld centres of $G$ form a subgroup of the group of
modular extensions. We identify this subgroup with a group of quadratic
extensions containing $G$ as a Lagrangian subgroup, the so-called group of
Lagrangian extensions of $G$. We compute the group of Lagrangian extensions,
thereby providing an interpretation of the internal structure of the third
cohomology group of an abelian $G$ in terms of fusion categories.
