  We investigate the high-scale behaviour of Higgs sectors beyond the Standard
Model, pointing out that the proper matching of the quartic couplings before
applying the renormalisation group equations (RGEs) is of crucial importance
for reliable predictions at larger energy scales. In particular, the common
practice of leading-order parameters in the RGE evolution is insufficient to
make precise statements on a given model's UV behaviour, typically resulting in
uncertainties of many orders of magnitude. We argue that, before applying
N-loop RGEs, a matching should even be performed at N-loop order in contrast to
common lore. We show both analytical and numerical results where the impact is
sizeable for three minimal extensions of the Standard Model: a singlet
extension, a second Higgs doublet and finally vector-like quarks. We highlight
that the known two-loop RGEs tend to moderate the running of their one-loop
counterparts, typically delaying the appearance of Landau poles. For the
addition of vector-like quarks we show that the complete two-loop matching and
RGE evolution hints at a stabilisation of the electroweak vacuum at high
energies, in contrast to results in the literature.
