  The axial form factor plays a crucial role in quasielastic neutrino-nucleus
scattering, but the error of the theoretical cross section due to uncertainties
of $G_A$ remains to be established. Reversely, the extraction of $G_A$ from the
neutrino nucleus cross section suffers from large systematic errors due to
nuclear model dependencies, while the use of single parameter dipole fits
underestimates the errors and prevents an identification of the relevant
kinematics for this determination. We propose to use a generalized
axial-vector-meson-dominance (AVMD) in conjunction with large-$N_c$ and high
energy QCD constrains to model the nucleon axial form factor, as well as the
half width rule as an a priori uncertainty estimate. The minimal hadronic
ansatz comprises the sum of two monopoles corresponding to the lightest
axial-vector mesons being coupled to the axial current. The parameters of the
resulting axial form factor are the masses and widths of the two axial mesons
as obtained from the averaged PDG values. By applying the half width rule in a
Monte Carlo simulation, a distribution of theoretical predictions can then be
generated for the neutrino-nucleus quasielastic cross section. We test the
model by applying it to the $(\nu_\mu,\mu)$ quasielastic cross section from
$^{12}$C for the kinematics of the MiniBooNE experiment. The resulting
predictions have no free parameters. We find that the relativistic Fermi gas
model globally reproduces the experimental data, giving $\chi^2/ \# bins =
0.81$. A $Q^2$-dependent error analysis of the neutrino data shows that the
uncertainties in the axial form factor $G_A(Q^2)$ are comparable to the ones
induced by the a priori half width rule. We identify the most sensitive region
to be in the range $0.2 \lesssim Q^2 \lesssim 0.6 \,{\rm GeV}^2$.
