  We present optimization of [(15 $\unicode{x212B}$) Ni$_{80}$Fe$_{20}$/(5
$\unicode{xC5}$) M]$_{20}$ single crystal multilayers on (001) MgO, with M
being Cu, Cu$_{50}$Pt$_{50}$ and Pt. These superlattices were characterized by
high-resolution X-ray reflectivity (XRR) and diffraction (XRD) as well as polar
mapping of important crystal planes. It is shown that cube on cube epitaxial
relationship can be obtained when depositing at the substrate temperature of
100 $^\circ$C regardless of the lattice mismatch (5% and 14% for Cu and Pt,
respectively). At lower substrate temperatures poly-crystalline multilayers
were obtained while at higher substrate temperatures {111} planes appear at
$\sim$10$^\circ$ off normal to the film plane. It is also shown that as the
epitaxial strain increases, the easy magnetization axis rotates towards the
direction that previously was assumed to be harder, i.e. from [110] to [100],
and eventually further increase in the strain makes the magnetic hysteresis
loops isotropic in the film plane. Higher epitaxial strain is also accompanied
with increased coercivity values. Thus, the effect of epitaxial strain on the
magnetocrystalline anisotropy is much larger than what was observed previously
in similar, but polycrystalline samples with uniaxial anisotropy (Kateb et al.
2021).
