Let \(G\) be a finite group, \(\mathbb ZG\) its integral group ring, \(\Delta\) its augmentation ideal, \(\Delta^n\) its powers, \(Q_n=\Delta^n/\Delta^{n+1}\) the \(n\)-th augmentation quotient as quotient of additive groups. We say that \(Q_n\) has the stable behaviour \(\{Q_i\mid n_0\leq i\leq n_0+\pi-1\}\) if for some least positive integers \(n_0\) and \(\pi\), \(Q_n\cong Q_{n+\pi}\) for all \(n\geq n_0\). By the classical result of F. Bachmann and L. Grünenfelder [J. Pure Appl. Algebra 5, 253-264 (1974; Zbl 0301.16011)], \(Q_n\) has the stable behaviour for nilpotent \(G\). The stable behaviour \(\{Q_i\mid n_0\leq i\leq n_0+\pi-1\}\) has already been described for finite abelian \(p\)-groups, for nonabelian \(G\) of order \(p\), \(p^2\), \(p^3\), \(p^4\) in a series of three papers, the third one is [K.-I. Tahara and T. Yamada, Jpn. J. Math., New Ser. 11, 109-130 (1985; Zbl 0584.20003)], \(2^5\) by G. Losey and N. Losey [Contemp. Math. 33, 412-435 (1984; Zbl 0556.20007)], for the semidihedral 2-group by J. Nan and H. Zhao [Czech. Math. J. 62, No. 1, 279-292 (2012; Zbl 1249.20004)].
There are 64 isomorphism classes of nonabelian groups \(G\) of order \(p^5\) for an odd prime \(p\), the author describes the stable behaviour \(\{Q_i\mid n_0\leq i\leq n_0+\pi-1\}\) of the augmentation quotients of \(\mathbb ZG\) in all these cases.