  In this note, we examine the gauging of the $\mathbb{Z}/2\mathbb{Z}$
permutation action on the tensor square of a modular tensor category. When
$\mathcal{C}$ has no nontrivial invertible objects, we provide formulas for the
fusion rules of both the extensions, expressed in terms of the fusion rules of
$\mathcal{C}$, and the subsequent equivariantizations, which additionally
requires the modular data of $\mathcal{C}$. We discuss several examples related
to quantum groups at roots of unity.
