  We analyse Starobinsky inflation in the presence of the Brout Englert Higgs
(BEH) boson with a non-minimal coupling to the Ricci scalar, $R$. The latter
induces a coupling of the massive scaleron associated with the $R^2$ term to
the BEH boson and this leads to a radiative correction to the BEH mass that
must be fine tuned to keep the scalar light. For the case of $R^2$ driven
inflation this requires a high level of fine tuning of order 1 part in
$10^{8}$; for the case of Higgs inflation it is very much greater. We consider
a scale invariant extension of the $R^2$/Higgs model and find that for $R^2$
driven inflation but not for Higgs inflation the required fine tuning is
significantly reduced to one part in $10^{3-4}$. We consider the vacuum
stability of the fine tuned model and its reheating and dilaton abundance after
inflation. We also discuss possible gravitational wave signals associated with
the model and the constraint on the mass of scalar or fermion dark matter
candidates if they are produced by the gravitational couplings of the scalaron.
