Infranormal Kazhdan subgroups and instability in finite actions

We prove that the group $G=\mathbb{Z}[1/2]^3\rtimes(\operatorname{SL}_3(\mathbb{Z})\times\mathbb{Z})$ is finitely presented and residually finite, but is not stable in finite actions, answering a question of Gohla and Thom. The proof uses the infranormal Kazhdan subgroup $H=\mathbb{Z}^3\rtimes\operatorname{SL}_3(\mathbb{Z})$ and its normal closure $N$ in $G$. A Kazhdan projection estimate shows that, in any $G$-action weakly contained in finite actions, every $H$-invariant measurable subset is also $N$-invariant. We construct an explicit sofic approximation of $G$ whose limit action contains a measurable subset that is $H$-invariant but not $N$-invariant.