Let \(L/K\) be a finite Galois extension of number fields with Galois group \(G\). Let \(n_G\) denote the \(\mathbb R\)-dimension of \(\bigcap_H\ker\text{Tr}_H\), where \(H\) ranges through the subgroups \(\neq1\) of \(G\) and \(\text{Tr}_H\) is the left \(G\)-endomorphism \(x\mapsto x\sum_{h\in H}h\) of the group ring \(\mathbb R[G]\). In Part I [Tohoku Math. J., II. Ser. 53, No. 1, 37--54 (2001; Zbl 0992.11061)], the authors computed the value of \(n_G\) for all groups \(G\) with the exception of two particular families. In the present paper they close the gap by showing that \(n_G=0\) for \(G= \text{SL}(2,\mathbb F_p)\), where \(p\) is a Fermat prime greater than \(5\).