  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 $B \cong B\otimes\mathcal{K}\otimes\mathcal{O}_2$ is
induced by an action on $B$. As a direct consequence, we establish that every
continuous action on a primitive ideal space 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 $\mathcal{O}_2$-stable
actions.
