  We calculate a class of three-loop Feynman diagrams which contribute to the
next-to-next-to-leading logarithmic approximation for the width difference
$\Delta\Gamma_{s}$ in the $B_s-\bar{B}_s$ system. The considered diagrams
contain a closed fermion loop in a gluon propagator and constitute the order
$\alpha_s^2 N_f$, where $N_f$ is the number of light quarks. Our results entail
a considerable correction in that order, if $\Delta\Gamma_{s}$ is expressed in
terms of the pole mass of the bottom quark. If the $\overline{MS}$ scheme is
used instead, the correction is much smaller. As a result, we find a decrease
of the scheme dependence. Our result also indicates that the usually quoted
value of the NLO renormalization scale dependence underestimates the
perturbative error.
