  Let $M$ be an irreducible projective variety over an algebraically closed
field $k$ of characteristic zero equipped with an action of a group $\Gamma$.
Let $E_G$ be a principal $G$--bundle over $M$, where $G$ is a connected
reductive algebraic group over $k$, equipped with a lift of the action of
$\Gamma$ on $M$. We give conditions for $E_G$ to admit a $\Gamma$--equivariant
reduction of structure group to $H$, where $H \subset G$ is a Levi subgroup. We
show that for $E_G$, there is a naturally associated conjugacy class of Levi
subgroups of $G$. Given a Levi subgroup $H$ in this conjugacy class, $E_G$
admits a $\Gamma$--equivariant reduction of structure group to $H$, and
furthermore, such a reduction is unique up to an automorphism of $E_G$ that
commutes with the action of $\Gamma$.
