  Let $M$ be an irreducible smooth projective variety, defined over an
algebraically closed field, equipped with an action of a connected reductive
affine algebraic group $G$, and let ${\mathcal L}$ be a $G$--equivariant very
ample line bundle on $M$. Assume that the GIT quotient $M/\!\!/G$ is a nonempty
set. We prove that the homomorphism of algebraic fundamental groups $\pi_1(M)\,
\longrightarrow\, \pi_1(M/\!\!/G)$, induced by the rational map $M\,
\longrightarrow\, M/\!\!/G$, is an isomorphism.
  If $k\,=\, \mathbb C$, then we show that the above rational map $M\,
\longrightarrow \, M/\!\!/G$ induces an isomorphism between the topological
fundamental groups.
