Continuous actions on primitive ideal spaces lift to $\mathrm{C}^{\ast}$-actions

Abstract: We prove that for any second-countable, locally compact group $G$, any continuous $G$-action on the primitive ideal space of a separable, nuclear $\mathrm{C}^{\ast}$-algebra $B$ such that $B \cong B\otimes\mathcal{K}\otimes\mathcal{O}_2$ is induced by an action on $B$. As a direct consequence, we establish that every continuous action on the primitive ideal space of a separable, nuclear $\mathrm{C}^{\ast}$-algebra is induced by an action on a $\mathrm{C}^{\ast}$-algebra with the same primitive ideal space. Moreover, we discuss an application to the classification of equivariantly $\mathcal{O}_2$-stable actions.