  In most iron-based superconductors, the transition to the magnetically
ordered state is closely linked to a lowering of structural symmetry from
tetragonal ($C_{4}$) to orthorhombic ($C_{2}$). However, recently, a regime of
$C_{4}$-symmetric magnetic order has been reported in certain hole-doped
iron-based superconductors. This novel magnetic ground state can be understood
as a double-$\mathbf{Q}$ spin density wave characterized by two order
parameters $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ related to each of the two
$\mathbf{Q}$ vectors. Depending on the relative orientations of the order
parameters, either a noncollinear spin-vortex crystal or a nonuniform
charge-spin density wave could form. Experimentally, M\"ossbauer spectroscopy,
neutron scattering, and muon spin rotation established the latter as the
magnetic configuration of some of these optimally hole-doped iron-based
superconductors. Theoretically, low-energy itinerant models do support a
transition from single-$\mathbf{Q}$ to double-$\mathbf{Q}$ magnetic order, but
with nearly-degenerate spin-vortex crystal and charge-spin density wave states.
In fact, extensions of these low-energy models including additional electronic
interactions tip the balance in favor of the spin-vortex crystal, in apparent
contradiction with the recent experimental findings. In this paper, we revisit
the phase diagram of magnetic ground states of low-energy multi-band models in
the presence of weak disorder. We show that impurity scattering not only
promotes the transition from $C_{2}$ to $C_{4}$-magnetic order, but it also
favors the charge-spin density wave over the spin-vortex crystal phase.
Additionally, in the single-$\mathbf{Q}$ phase, our analysis of the nematic
coupling constant in the presence of disorder supports the experimental finding
that the splitting between the structural and stripe-magnetic transition is
enhanced by disorder.
