tracially strictly approximately inner action.

The Cuntz semigroup is a key ingredient in the Elliott program for the classification of C\*-algebras [@ET08; @Tom08]. We refer to [@APT11; @TT15] for many aspects of the Cuntz semigroup. It is generally large and complicated. Among simple nuclear [C\*-algebra]{}[s]{}, the classifiable ones are those whose Cuntz semigroups are easily accessible (Section 5 of [@APT11]). With the near completion of the Elliott program, nonclassifiable [C\*-algebra]{}[s]{} attract more attention (see [@AGP19; @HP19; @Ph16]) and the Cuntz semigroup is the main additional available invariant. The goal of this paper is to conduct an investigation beyond classifiable C\*-algebras by considering crossed products by finite groups of simple C\*-algebras which may not have strict comparison of positive elements, although there are still many pieces of the puzzle that have to be put together.

Section [\[Sec\_Prelim\]]{} is devoted to providing a quick overview of the background we need here. In Section \[Sec\_Approx\_Innner\] and Section \[Sec\_Tracial\_Approx\_Innner\], we give a standard definition for a strictly approximately inner action of a finite group on a unital C\*-algebra and its tracial analog, tracial strict approximate innerness. (See Definition \[D\_9719\_StrictAppInn\] and Definition \[D\_9731\_TrStrAppInn\].) To provide these definitions, we basically give a generalization of the Definition 