Comparing Two Notions of Coaction Invariance of Ideals in $\mathrm{C}^*$-Algebras

Given a coaction $\delta$ of a locally compact group $G$ on a $\mathrm{C}^*$-algebra $A$, we study the relationship between two different forms of coaction invariance of ideals of $A$ and the ideals of the corresponding crossed product $\mathrm{C}^*$-algebra $A \rtimes_{\delta} G$.

In particular, we characterize when these two notions of invariance are equivalent.