Two-loop QED corrections with closed fermion loops for the bound-electron g factor

Two-loop QED corrections with closed fermion loops are calculated for the $1s$ bound-electron g factor. Calculations are performed to all orders in the nuclear binding strength parameter Zα (where Z is the nuclear charge and α is the fine-structure constant) except for the closed fermion loop, which is treated within the free-loop (Uehling) approximation in some cases. Comparison with previous Zα-expansion calculations is made and the higher-order remainder of order α²(Zα)⁵ and higher is separated out from the numerical results.