The core of a weakly group-theoretical braided fusion category

We show that the core of a weakly group-theoretical braided fusion category $C$ is equivalent as a braided fusion category to a tensor product $B boxtimes D$, where $D$ is a pointed weakly anisotropic braided fusion category, and $B cong vect$ or $B$ is an Ising braided category. In particular, if $C$ is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius-Perron dimension at most 2 is necessarily group-theoretical.