effects in strong fields. Accurate measurements of the groun d-state hyperfine splittings were per- formed for several H-like ions, including $^{209}$Bi, $^{165}$Ho, $^{185}$Re, $^{187}$Re, $^{207}$Pb, $^{203}$Tl, and $^{205}$Tl in Refs. [28–32]. These experiments motivated corresponding theor etical investigations [33–40]. The the- oretical uncertainty of the hyperfine splitting is also domi nated by the nuclear effects, mainly by the Bohr-Weisskopf effect [35]. The theoretical investigations have shown [40] that simultaneous studies of the hyperfine splitting in H-like and Li-like ionsof the same isotope can significantly improve the accuracy of the theoretical prediction. As a result, the ground-state hyperfine splitting in Li-like bismuth was predicted to a high accuracy using the experimental result for the 1 s hy- perfine splitting in H-like bismuth [38–40]. The indirect me asurement of the hyperfine splitting of lithiumlike bismuth performed in Livermore yielded value of 820(26) meV [41]. Determination of this splitting to a much higher accuracy (of about 10 − $^{7}$) is planned at GSI in the framework of the HITRAP project [42]. This requires further improvements of the theoretical predictions for Li-like ions and, in particular, evaluations of the QED screening effect. An approximate treatment of this effect was accomplished in Refs. [39, 43–47] by emplo ying an effective screening potential in calculations of the one-loop self-energy and vacuum-polarization diagrams. Rigorous evaluation of the two-electron self-energy and vacuum-polarization diagrams (Figs. 1 and 2) remained a challenge for theory until recently. In our Letter [48] the complete αZ - dependent contributions of the two-electron self-energy diagrams and a dominant part of the two- electron vacuum-polarization diagrams have been evaluated for the hyperfine structure of Li-like bismuth and for the g factor of Li-like lead. In the present paper we describe in detail the evalua- tion of the screened quantum electrodynamical correction in presence of external magnetic field. Furthermore, we extend our calculations to the wide range of nuclear charge Z = 20 – 83 in case of the hyperfine splitting. The accuracy of the theoretical p rediction for the specific difference of the hyperfine splittings of H- and Li-like bismuth is improve d. As to the g factor, we present the results for lead and uranium ions. The relativistic units ( ℏ = 1 , c = 1 , m = 1 ) and the Heaviside charge unit [ α = e $^{2}$/ (4 π ) , e < 0 ] are used throughout the paper. 3