  We prove the universality of predictions for linear perturbations from the
entire class of models of inflation driven by a pseudo-scalar field coupled to
an SU($N$) gauge boson, where SU($2$) subgroups in the SU($N$) crossed with the
background spatial SO($3$) spontaneously break into a single SO($3$). The
effect of which SU($2$) subgroup in SU($N$) acquires a VEV through spontaneous
symmetry breaking can be quantified by a single parameter $\lambda$, which
always appears in combination with the gauge coupling constant $g$. In the
linear perturbations, as well as the background system, the same dynamics and
predictions as in the chromo-natural inflation hold for its SU($N$) extension
by replacing $g \to g\lambda$. The latter models thereby draw the same
prediction curve on the $n_s$-$r$ plane as the former at the tree level as long
as $g \lambda$ stays constant during inflation. We briefly discuss possible
transitions from one value of $\lambda$ to another during inflation and the
observational prospects.
