The exposed points of the set of invariant means on an ideal

Let $G$ be a $\sigma$-compact locally compact nondiscrete group and let $Q$ be a $G$-invariant ideal of $L^{\infty }(G)$. We denote the set of left invariant means $m$ on $L^{\infty }(G)$ that are zero on $Q$ (i.e. $m(f) = 0$ for all $f\in Q$) by $LIM_{Q}$. We show that, when $G$ is amenable as a discrete group and the closed $G$-invariant subset of the spectrum of $L^{\infty }(G)$ corresponding to $Q$ is a $G_{\delta }$-set, $LIM_{Q}$ is very large in the sense that every nonempty $G_{\delta }$-subset of $LIM_{Q}$ contains a norm discrete copy of $\beta \mathbb{N}$, where $\beta \mathbb{N}$ is the Stone-$\mathrm{\check{C}ech}$ compactification of the set $\mathbb{N}$ of positive integers with the discrete topology. In particular, we prove that $LIM_{Q}$ has no exposed points in this case and every nonempty $G_{\delta }$-subset of the set of left invariant means on $L^{\infty }(G)$ contains a norm discrete copy of $\beta \mathbb{N}$.