We investigated head-to-head domain walls in nanostrips of epitaxial $\mathrm{Fe}_4\mathrm{N}(001)$ thin films, displaying a fourfold magnetic anisotropy. Magnetic force microscopy and micromagnetic simulations show that the domain walls have specific properties, compared to soft magnetic materials. In particular, strips aligned along a hard axis of magnetization are wrapped by partial flux-closure concertina domains below a critical width, while progressively transforming to zigzag walls for wider strips. Transverse walls are favored upon initial application of a magnetic field transverse to the strip, while transformation to a vortex walls is favored upon motion under a longitudinal magnetic field. In all cases the magnetization texture of such fourfold anisotropy domain walls exhibits narrow micro-domain walls, which may give rise to peculiar spin-transfer features.