Reading about induced Systems of $C^*$-Algebras, I found this one statement that I can't figure out.

Let $G$ be a compact Group and $(A,G,\alpha)$ a $C^*$-dynamical System, such that for some closed subgroup $H$ of $G$ there exists a $G$-equivariant map $\varphi$ between $(\text{Prim}(A),G,\alpha)$ and $(G/H,G,\text{lt})$, where lt denotes the left-translation. Let $I:= \bigcap \{P\in \text{Prim}(A) \colon \varphi(P)=eH\}$.

Now it is stated that $\bigcap\{\alpha_s(I) : s\in G\}$ equals $\{0\}$. I don't really get. I know that $I$ is an $H$-invariant ideal in $A$, since $\varphi$ is equivariant and all the $\alpha_s$ are *-automorphisms.

Thank you in advance!

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