  Nonperturbative lattice field theory simulations provide a systematic
framework to investigate properties of conformal systems at strong couplings.
These simulations can be performed using different lattice discretizations.
Here we present numerical results for the step scaling beta function in SU(3)
gauge theories with ten and twelve fundamental flavors. We calculate the
renormalized $\beta$ function in the finite volume gradient flow
renormalization scheme. Using M\"obius domain wall fermions with Symanzik gauge
action, Zeuthen gradient flow, and perturbative tree-level improvement, we
implement a fully $O(a^2)$ Symanzik improved set-up and demonstrate its
advantages. We compare our findings to existing results in the literature. For
the ten flavor system we observe excellent agreement with the domain wall
step-scaling function calculated by Chiu for the range in $g_c^2$ where our
data overlap. In the case of the twelve flavor system, our $O(a^2)$ Symanzik
improved set-up predicts a conformal infrared fixed point around $g_c^2 \sim
5.5$ in the $c=0.25$ scheme, which is presently in tension with staggered
fermion results in the literature. We consider possible reasons for the
discrepancy.
