  We report calculations of QED corrections to the $g$ factor of Li-like ions
induced by the exchange of two virtual photons between the electrons. The
calculations are performed within QED theory to all orders in the nuclear
binding strength parameter $Z\alpha$, where $Z$ is the nuclear charge number
and $\alpha$ is the fine-structure constant. In the region of low nuclear
charges we compare results from three different methods: QED, relativistic
many-body perturbation theory, and nonrelativistic QED. All three methods are
shown to yield consistent results. With our calculations we improve the
accuracy of the theoretical predictions of the $g$ factor of the ground state
of Li-like carbon and oxygen by about an order of magnitude. Our theoretical
results agree with those from previous calculations but differ by 3-4 standard
deviations from the experimental results available for silicon and calcium.
