  Let $W$ be a finite dimensional purely odd supervector space over
$\mathbb{C}$, and let $\sRep(W)$ be the finite symmetric tensor category of
finite dimensional superrepresentations of the finite supergroup $W$. We show
that the set of equivalence classes of finite non-degenerate braided tensor
categories $\C$ containing $\sRep(W)$ as a Lagrangian subcategory is a torsor
over the cyclic group $\mathbb{Z}/16\mathbb{Z}$. In particular, we obtain that
there are $8$ non-equivalent such braided tensor categories $\C$ which are
integral and $8$ which are non-integral.
