  There are many hints that gravity is asymptotically safe. The inclusion of
gravitational corrections can result in the ultraviolet fundamental Standard
Model and constrain the Higgs mass to take the smallest value such that
electroweak vacuum is stable. Taking into account the current top quark mass
measurements this value is $m_H \approx 130$ GeV.\\ This article considers the
predictions of the Higgs mass in two minimal Beyond Standard Model scenarios,
where the stability is improved. One is the sterile quark axion model, while
the other is the $U(1)_{B-L}$ gauge symmetry model introducing a new massive
$Z'$ gauge boson. The inclusion of $Z'$ boson gives the correct prediction for
this mass, while inclusion of sterile quark(s) gives only a slight effect.\\
Also a new, gravitational solution to the strong CP problem is discussed.
