  It is known that the category $\mathscr{C}^H$ of nilpotent weight modules
over the quantum group associated with the super Lie algebra
$\mathfrak{sl}(2|1)$ is a relative pre-modular $G$-category. Its modified trace
enables to define an invariant of $3$-manifolds. In this article we show that
the category $\mathscr{C}^H$ is a relative modular $G$-category which allows
one to construct a family of non-semi-simple extended topological quantum field
theories which surprisingly are anomaly free. The quantum group associated with
$\mathfrak{sl}(2|1)$ is considered at odd roots of unity.
