${\mathrm{C}^*}$-algebras, Izumi  introduced Rokhlin property of finite group actions on ${\mathrm{C}^*}$-algebras extracting an essential property which is used to classify group actions in the von Neumann algebra theory and classified such actions. He also shows that many of fundamental ${\mathrm{C}^*}$-dynamical systems such as infinite tensor products of left regular actions of a finite group $G$ on $\mathbb{M} _{|G|}$ and the quasi-free action of a finite group $G$ on ${\mathcal{O}}_{|G|}$ by a regular representation has this property. After his work, the study of this kind of actions has been attracting attention among ${\mathrm{C}^*}$-algebraists . Among those, recently, Gardella [@mathOA14081946] initiated the study of Rokhlin actions for general compact groups.

Roughly speaking, Rokhlin property is an analogue in an approximate sense of freeness of product type (that is, $G$-actions on $G \times X$) in topological dynamics. For a $G$-${\mathrm{C}^*}$-algebra $A$ with Rokhlin property, there is a fundamental technique replacing a projection or a unitary with another $G$-invariant one, which is called an *averaging technique* by Gardella [@mathOA14075485]. More precisely, there is a sequence of completely positive maps $A \to A^\alpha$ which is approximately a $\ast$-homomorphism preserving $A^\alpha$. This gives a strong restriction on the structure of ${\mathrm{C}^*}$-dynamical systems, its ${\mathrm{K}}$-groups, crossed products 