Back in the 90's, Nilsen proved the following result for normal $C^{*}$-dynamical systems:
Suppose that $A$ is a $C^{*}$-algebra, and $G$ is a locally compact group.
Let $\delta$ be a coaction of $G$ on $A$. Then restriction gives a bijection between the $\hat{\delta}$-invariant ideals of $A\rtimes_\delta G$ and the $\delta$-invariant ideals of $A$.
Let $\alpha$ be an action of $G$ on $A$. Then restriction gives a bijection between the $\hat{\alpha}$-invariant ideals of $A\rtimes_\alpha G$ and the $\alpha$-invariant ideals of $A$.
This establishes a bijective correspondence between the action and dual coaction invariant ideals of $A$ and its crossed product by $G$. My question is that if we have a system $(A,G,\alpha)$ twisted by a closed normal subgroup $N \lhd G$, does such an analogous result exist?
For instance, if $J \subset A \rtimes_{\alpha|} N$ is an ideal which is $\alpha_N$-invariant ($\alpha_N$ the twisted action), is there a result that states that $J$ must be of the form $J = I \rtimes_{\beta} N$ for $\beta$ some altered version of the action $\alpha$ (or perhaps something entirely different)?
I'm aware of a much more restrictive result of a similar nature from this paper, but I haven't been able to find a direct analog to Nilsen's results to systems twisted by closed normal subgroups $N \lhd G$.
Any help would be appreciated, thank you.