Let \(\mathfrak X\) be a class of groups. A group \(G\) is called just-non-\(\mathfrak X\) if it is not in the class \({\mathfrak X}\) but all its proper quotients are \(\mathfrak X\)-groups. The structure of just-non-\(\mathfrak X\) has been investigated for several group classes \(\mathfrak X\) (see for instance \textit{J. S. Wilson} [Proc. Camb. Philos. Soc. 69, 373--391 (1971; Zbl 0216.08803)], \textit{D. J. S. Robinson} and \textit{J. S. Wilson} [Proc. Lond. Math. Soc. (3) 48, 193-229 (1984; Zbl 0526.20025)], \textit{S. Franciosi} and \textit{F. de Giovanni} [Atti Accad. Naz. Lincei, III. Ser. Rend. Cl. Sci. Fis. Mat. Nat. 79, 19--24 (1985; Zbl 0639.20020)], \textit{F. de Giovanni} [Boll. Unione Mat. Ital., VII. Ser. B 5, 449--462 (1991; Zbl 0744.20032)]). Clearly the class of just-non-\(\mathfrak X\) groups is contained in the class \(\mathfrak X\) of all groups \(G\) such that \(G\not\in {\mathfrak X}\) but \(G/N\) is an \(\mathfrak X\)-group for every normal subgroup \(N\) of \(G\) which is not in \(\mathfrak X\). In this context infinite groups in which every infinite normal subgroup has finite index were already considered by the reviewer [Ric. Mat. 38, 151--163 (1989; Zbl 0726.20025)]. Here the authors investigate the class \(\widetilde{\mathfrak M}\), where \(\mathfrak M\) is the class of minimax groups, and in particular they obtain information on the structure of \(\widetilde {\mathfrak M}\)-groups having an ascending normal series whose factors either are finite or abelian minimax groups.