  The familiar Fourier-Mukai technique can be extended to an equivariant
setting where a finite group $G$ acts on a smooth projective variety $X$. In
this paper we compare the group of invariant autoequivalences $\Aut(D(X))^G$
with the group of autoequivalences of $D^G(X)$. We apply this method in three
cases: Hilbert schemes on K3 surfaces, Kummer surfaces and canonical quotients.
