Automorphisms of the Doubles of Purely Non-abelian Finite Groups

Using a recent classification of End(D(G)), we determine a number of properties for Aut(D(G)), where D(G) is the Drinfel'd double of a finite group G. Furthermore, we completely describe Aut(D(G)) for all purely non-abelian finite groups G. A description of the action of Aut(D(G)) on Rep(D(G)) is also given. We are also able to produce a simple proof that D(G) congruent to D(H) if and only if G congruent to H, for G and H finite groups.