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We report the results of a lattice quantum chromodynamics calculation of$F_K/F_\pi$ using M\"{o}bius domain-wall fermions computed on gradient-flowed$N_f=2+1+1$ highly-improved staggered quark (HISQ) ensembles. The calculationis performed with five values of the pion mass ranging from $130 \lesssim m_\pi\lesssim 400$ MeV, four lattice spacings of $a\sim 0.15, 0.12, 0.09$ and $0.06$fm and multiple values of the lattice volume. The interpolation/extrapolationto the physical pion and kaon mass point, the continuum, and infinite volumelimits are performed with a variety of different extrapolation functionsutilizing both the relevant mixed-action effective field theory expressions aswell as discretization-enhanced continuum chiral perturbation theory formulas.We find that the $a\sim0.06$ fm ensemble is helpful, but not necessary toachieve a subpercent determination of $F_K/F_\pi$. We also include an estimateof the strong isospin breaking corrections and arrive at a final result of$F_{K^\pm}/F_{\pi^\pm} = 1.1942(45)$ with all sources of statistical andsystematic uncertainty included. This is consistent with the Flavour LatticeAveraging Group average value, providing an important benchmark for our latticeaction. Combining our result with experimental measurements of the pion andkaon leptonic decays leads to a determination of $|V_{us}|/|V_{ud}| =0.2311(10)$.

$F_K / F_\pi$ from M\"{o}bius domain-wall fermions solved on gradient-flowed HISQ ensembles