Certain tracially nuclear dimensional for certain crossed product ${\mathrm{C}}^{\ast }$-algebras

Let $\mathrm{\Omega }$ be a class of unital ${\mathrm{C}}^{\ast }$-algebras which have the second type tracial nuclear dimensional at moat $n$ (or have tracial nuclear dimensional at most $n$). Let $A$ be an infinite dimensional unital simple ${\mathrm{C}}^{\ast }$-algebra such that $A$ is asymptotical tracially in $\mathrm{\Omega }$. Then ${\mathrm{T}}^2\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{n}\mathrm{u}\mathrm{c}}(A)\le n$ (or $\mathrm{T}\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{n}\mathrm{u}\mathrm{c}}(A)\le n$). As an application, let $A$ be an infinite dimensional simple separable amenable unital ${\mathrm{C}}^{\ast }$-algebra with ${\mathrm{T}}^2\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{n}\mathrm{u}\mathrm{c}}(A)\le n$ (or $\mathrm{T}\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{n}\mathrm{u}\mathrm{c}}(A)\le n$). Suppose that $\alpha :G\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$ is an action of a finite group $G$ on $A$ which has the tracial Rokhlin property. Then ${\mathrm{T}}^2\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{n}\mathrm{u}\mathrm{c}}({\mathrm{C}}^{\ast }(G,A,\alpha ))\le n$ (or $\mathrm{T}\mathrm{d}\mathrm{i}{\mathrm{m}}_{\mathrm{n}\mathrm{u}\mathrm{c}}$ $({\mathrm{C}}^{\ast }(G,A,\alpha ))\le n$).