  We study the relationships between the basic parameters of the on-shell
renormalization scheme and their counterparts in the $\overline{\mathrm{MS}}$
scheme at full order ${\cal O}(\alpha^2)$ in the Standard Model. These enter as
threshold corrections the renormalization group analyses underlying, e.g., the
investigation of the vacuum stability. To ensure the gauge invariance of the
parameters, in particular of the $\overline{\mathrm{MS}}$ masses, we work in
$R_\xi$ gauge and systematically include tadpole contributions. We also
consider the gaugeless-limit approximation and compare it with the full
two-loop electroweak calculation.
