  We present an approach of calculating the group of braided autoequivalences
of the category of representations of the Drinfeld double of a finite
dimensional Hopf algebra $H$ and thus the Brauer-Picard group of
$H$-$\mathrm{mod}$. We consider two natural subgroups and a subset as
candidates for generators. In this article $H$ is the group algebra of a finite
group $G$. As our main result we prove that any element of the Brauer-Picard
group, fulfilling an additional cohomological condition, decomposes into an
ordered product of our candidates. For elementary abelian groups $G$ our
decomposition reduces to the Bruhat decomposition of the Brauer-Picard group,
which is in this case a Lie group over a finite field. Our results are
motivated by and have applications to symmetries and defects in $3d$-TQFT and
group extensions of fusion categories.
