  We commence the evaluation of the one- and two-loop binding corrections to
the $g factor for an electron in a hydrogenlike system of order alpha^2 (Z
alpha)^5 and consider diagrams with closed fermion loops. The one-loop
vacuum-polarization correction is rederived and confirmed. For the two-loop
vacuum-polarization correction, due to a specific gauge-invariant set of
diagrams with closed fermion loops, we find a correction delta g = 7.442
(alpha/pi)^2 (Z alpha)^5. Based on the numerical trend of the coefficients
inferred from the gauge-invariant subset, we obtain a numerically large
tentative estimate for the complete two-loop binding correction to the g factor
(sum of self-energy and vacuum polarization).
