Let \({\mathcal G}= \langle G;+,-,0 \rangle\) be an elementary 2-group. Let \(F_G\) be the set of all \({\mathcal G}\)-affine (linear) \(n\)-ary functions \(f(x_1,\dots, x_n)= g+ x_{i_1}+\dots +x_{i_t}\) \((g\in G\), \(1\leq i_1<\dots< i_t\leq n)\) on \(G\) and let \(D_{\mathcal G}\) be the set of all idempotent (i.e. \(f(g,\dots, g)= g\) for all \(g\in G\)) functions \(f\in F_{\mathcal G}\). Let \(f\) be an \(n\)-ary function on \(G\), \(g> 3\); let \(f_{ij} (x_1,\dots, x_{n-1})= f(x_1,\dots, x_{j-1}, x_i, x_j,\dots, x_{n-1})\), \(f_i^g (x_1,\dots, x_{n-1})= f(x_1,\dots, x_{i-1},g, x_i,\dots, x_{n-1})\). It is proved that: 1) if all functions \(f_{ij}\) obtained from \(f\) are \({\mathcal G}\)-affine then there exists an \(n\)-ary \({\mathcal G}\)-affine function \(h\) on \(G\) such that \(f(g_1,\dots, g_n)= h(g_1,\dots, g_n)\) whenever some of \(g_1,\dots, g_n\in G\) are equal; 2) if all \(f_{ij}, f_i^g\in D_{\mathcal G}\), then \(n\) is odd and \(f(g_1,\dots, g_n)= g_1+\dots+ g_n\) whenever some of \(g_1,\dots, g_n\in G\) are equal.