Two-loop QED corrections with closed fermion loops

We report a calculation of all two-loop quantum electrodynamics corrections with closed fermion loops for the $n=1$ and $n=2$ states of H-like ions and for a wide range of the nuclear charge numbers $Z=1\char21{}100$. The calculation is performed to all orders in the binding-strength parameter $Z\ensuremath{\alpha}$, with the exception that in a few cases the free-loop approximation is employed in the treatment of the fermion loops. Detailed comparison is made with previous $Z\ensuremath{\alpha}$-expansion calculations and the higher-order remainder term to order ${\ensuremath{\alpha}}^{2}{(Z\ensuremath{\alpha})}^{6}$ is identified.