  Let $A$ be a separable, unital, simple, $\mathcal{Z}$-stable, nuclear
$C^*$-algebra, and let $\alpha\colon G\to \mathrm{Aut}(A)$ be an action of a
discrete, countable, amenable group. Suppose that the orbits of the action of
$G$ on $T(A)$ are finite and that their cardinality is bounded. We show that
$\alpha$ is strongly outer if and only if
$\alpha\otimes\mathrm{id}_{\mathcal{Z}}$ has the weak tracial Rokhlin property.
If $G$ is moreover residually finite, these conditions are also equivalent to
$\alpha\otimes\mathrm{id}_{\mathcal{Z}}$ having finite Rokhlin dimension (in
fact, at most 2).
  If $\partial_eT(A)$ is furthermore compact, has finite covering dimension,
and the orbit space $\partial_eT(A)/G$ is Hausdorff, we generalize results by
Matui and Sato to show that $\alpha$ is cocycle conjugate to
$\alpha\otimes\mathrm{id}_{\mathcal{Z}}$, even if $\alpha$ is not strongly
outer. In particular, in this case the equivalences above hold for $\alpha$ in
place of $\alpha\otimes\mathrm{id}_{\mathcal{Z}}$. In the course of the proof,
we develop equivariant versions of complemented partitions of unity and uniform
property $\Gamma$ as technical tools of independent interest.
