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

We prove that for any second-countable, locally compact group $G$, any continuous $G$-action on the primitive ideal space of a separable, nuclear $C^{*}$-algebra $B$ such that $B\cong B\otimes O_2\otimes K$ 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 $C^{*}$-algebra is induced by an action on a $C^{*}$-algebra with the same primitive ideal space. Moreover, we discuss an application to the classification of equivariantly $O_2$-stable actions.