  We present a numerical evaluation of the loop-after-loop contribution to the
second-order self-energy for the ground state of hydrogenlike atoms with low
nuclear charge numbers Z. The calculation is carried out in the Fried-Yennie
gauge and without an expansion in Z \alpha. Our calculation confirms the
results of Mallampalli and Sapirstein and disagrees with the calculation by
Goidenko and coworkers. A discrepancy between different calculations is
investigated. An accurate fitting of the numerical results provides a detailed
comparison with analytic calculations based on an expansion in the parameter Z
\alpha. We confirm the analytic results of order \alpha^2 (Z\alpha)^5 but
disagree with Karshenboim's calculation of the \alpha^2 (Z \alpha)^6 \ln^3(Z
\alpha)^{-2} contribution.
