K. Ikegami and O. Saeki [J. Math. Soc. Japan 55, No.4, 1081-1094 (2003; Zbl 1046.57020)] introduced a notion of cobordism for Morse functions. Two Morse functions \(f_0: M_0\to \mathbb R\) and \(f_1: M_1\to \mathbb R\) on closed \(n\)-dimensional manifolds \(M_0\) and \(M_1\) are cobordant if there exists a compact \((n+1)\)-dimensional manifold \(X\) with \(\partial X= M_0\sqcup M_1\) and a smooth map \(F: X\to {\mathbb R}\times [0,1]\) extending \(f_0\sqcup f_1\) and having only fold singularities. In certain contexts the manifold \(X\) is required to have additional properties. K. Ikegami and O. Saeki proved that the cobordism group of Morse functions on oriented surfaces is infinite cyclic. The result of the paper under review states that the cobordism group of Morse functions on nonorientable surfaces is \({\mathbb{Z}}\oplus {\mathbb Z}_2\). The problem of calculating the cobordism group of Morse functions on manifolds was solved in general by K. Ikegami [Hiroshima Math. J. 34, No.2, 211-230 (2004; Zbl 1064.57037)].