13.59 (1995)

One can show that any extension of shape $\mathbb{Z}^d \cdot SL_d(\mathbb{Z})$ is residually finite, unless possibly $d = 3$ or $d = 5$. Are there in fact any extensions of this shape that fail to be residually finite when $d = 5$? As shown in (P. R. Hewitt, Groups/St. Andrews '93 in Galway, Vol. 2 (London Math. Soc. Lecture Note Ser., 212), Cambridge Univ. Press, 1995, 305–313), there is an extension of $\mathbb{Z}^3$ by $SL_3(\mathbb{Z})$ that is not residually finite. Usually it is true that an extension of an arithmetic subgroup of a Chevalley group over a rational module is residually finite. Is it ever false, apart from the examples of shape $\mathbb{Z}^3 \cdot SL_3(\mathbb{Z})$?