Homotopy types of $\mathrm{Spin}^c(n)$-gauge groups over $S^4$

The gauge group of a principal $G$-bundle $P$ over a space $X$ is the group of $G$-equivariant homeomorphisms of $P$ that cover the identity on $X$. We consider the gauge groups of bundles over $S^4$ with $\mathrm{Spin}^c(n)$, the complex spin group, as structure group and show how the study of their homotopy types reduces to that of $\mathrm{Spin}(n)$-gauge groups over $S^4$. We then advance on what is known by providing a partial classification for $\mathrm{Spin}(7)$- and $\mathrm{Spin}(8)$-gauge groups over $S^4$.