Unique equivariant pseudo expectation and generalized Powers averaging for crossed product

Let $\mathrm{\Gamma }$ be a countable discrete group and let $A$ be a unital $\mathrm{\Gamma }$-simple $\mathrm{\Gamma }$-$C^{\ast }$-algebra. We prove that $A{⋊}_r\mathrm{\Gamma }$ has the generalized Powers averaging property if and only if the canonical map $A{⋊}_r\mathrm{\Gamma }\to I_{\mathrm{\Gamma }}(A)$ is the unique $\mathrm{\Gamma }$-equivariant pseudo-expectation. We also show that this averaging property lifts to $I_{\mathrm{\Gamma }}(A){⋊}_r\mathrm{\Gamma }$, with the averaging coefficients still belonging to $A$.