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-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. (C) 2017 Elsevier Inc. All rights reserved.