  We examine the relation between the gauge groups of $\mathrm{SU}(n)$- and
$\mathrm{PU}(n)$-bundles over $S^{2i}$, with $2\leq i\leq n$, particularly when
$n$ is a prime. As special cases, for $\mathrm{PU}(5)$-bundles over $S^4$, we
show that there is a rational or $p$-local equivalence
$\mathcal{G}_{2,k}\simeq_{(p)}\mathcal{G}_{2,l}$ for any prime $p$ if, and only
if, $(120,k)=(120,l)$, while for $\mathrm{PU}(3)$-bundles over $S^6$ there is
an integral equivalence $\mathcal{G}_{3,k}\simeq\mathcal{G}_{3,l}$ if, and only
if, $(120,k)=(120,l)$.
