  We extensively motivate the studies of higher-derivative gravities, and in
particular we emphasize which new quantum features theories with six
derivatives in their definitions possess. Next, we discuss the mathematical
structure of the exact on the full quantum level beta functions obtained
previously for three couplings in front of generally covariant terms with four
derivatives (Weyl tensor squared, Ricci scalar squared and the Gauss-Bonnet
scalar) in minimal six-derivative quantum gravity in $d=4$ spacetime
dimensions. The fundamental role here is played by the ratio $x$ of the
coupling in front of the term with Weyl tensors to the coupling in front of the
term with Ricci scalars in the original action. We draw a relation between the
polynomial dependence on $x$ and the absence/presence of enhanced conformal
symmetry and renormalizability in the models where formally $x\to+\infty$ in
the case of four- and six-derivative theories respectively.
