The author studies some properties of the prime ideals in the group ring \(kG\), where \(k\) is a finite field and \(G\) is an Abelian minimax group. Recall that an Abelian group \(G\) is minimax if \(G\) contains a finitely generated subgroup \(H\) such that \(G/H\) satisfies the minimal condition. An Abelian minimax group \(G\) is called reduced if the torsion subgroup of \(G\) is finite. Let \(Q\) be a prime ideal of \(kG\) and suppose that the factor group \(G/(1+Q)\cap G\) is reduced. Let \(L(Q)\) denote the set of all ideals \(L\) of \(kG\) such that \(Q\subseteq L\) and \(L\) is a maximal ideal of finite index in \(kG\). Then the intersection of all ideals \(L\in L(Q)\) coincides with \(Q\). This theorem generalizes the main result of part I [\textit{D. Segal}, J. Algebra 237, No. 1, 64-94 (2001; Zbl 0989.20003)]. Let \(\Gamma\) be a group, acting by automorphisms on \(G\) and let \(P\) be a nonzero \(\Gamma\)-invariant prime ideal of \(kG\). If \((1+ P)\cap G=1\), then \(G\) has a \(\Gamma\)-invariant subgroup \(H\) such that \(H\) controls \(P\), i.e., \(P=(P\cap kH)kG\), and for some subgroup \(\Gamma_0\) of finite index in \(\Gamma\) the groups \(H\) and \(\Gamma_0\) form a so-called strict Brookes pair. Moreover, for a prime ideal \(Q\) of \(kG\) and fixed element \(\lambda\in kG\) let \(L(\Gamma,Q,\lambda)\) denote the set of all ideals \(L\) of \(kG\) such that \(Q\subseteq L\), \(L\) is a maximal ideal of finite index in \(kG\) and \(\lambda^\gamma\not\in L\) for all \(\gamma\in\Gamma\). If \(\Gamma\) is a virtually soluble group and \(G/(1+Q)\cap G\) is reduced, then the intersection of all ideals \(L\in L(\Gamma,Q,\lambda)\) coincides with \(Q\). This result generalizes the main Theorem 2.1 of \textit{D. Segal} [Trans. Am. Math. Soc. 353, No. 1, 391-410 (2001; Zbl 0959.20004)].