  We show that if $A$ is a simple (not necessarily unital) tracially
$\mathcal{Z}$-absorbing C*-algebra and $\alpha \colon G \to \mathrm{Aut} (A)$
is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin
property, then the crossed product $C^*(G, A,\alpha)$ and the fixed point
algebra $A^\alpha$ are simple and tracially $\mathcal{Z}$-absorbing, and they
are $\mathcal{Z}$-stable if, in addition, $A$ is separable and nuclear. The
same conclusion holds for all intermediate C*-algebras of the inclusions
$A^\alpha \subseteq A$ and $A \subseteq C^*(G, A,\alpha)$. We prove that if $A$
is a simple tracially $\mathcal{Z}$-absorbing C*-algebra, then, under a
finiteness condition, the permutation action of the symmetric group $S_m$ on
the minimal $m$-fold tensor product of $A$ has the weak tracial Rokhlin
property. We define the weak tracial Rokhlin property for automorphisms of
simple C*-algebras and we show that -- under a mild assumption -- (tracial)
$\mathcal{Z}$-absorption is preserved under crossed products by such
automorphisms.
