Let \(\ell\) be an odd prime and let \(p\neq\ell\) be a prime which generates the \(\ell\)-adic units. Let \(\zeta_ a\) be a primitive \(\ell^ a\)-th root of unity and let \(\mu\) be the group of \(\ell\)-primary roots of unity in \(\mathbb{Z}[\zeta_ a]\). Then there is a natural map \(h: Q_ 0(B\mu_ +)\to\text{BGL}(\mathbb{Z}[\zeta_ a])^ +\), where \(B\mu_ +\) is the classifying space of \(\mu\) with an added base point, \(Q_ 0(\;)\) denotes the infinite loop space associated to the suspension spectrum and the superscript \(+\) denotes Quillen's plus construction. In \(\mathbb{Z}[\zeta_ a]\) there is a unique prime above \(p\) with residue class field \(\mathbb{F}_ q:=\mathbb{F}_ p[\zeta_ a]\). Generalizing an earlier result of Quillen, it was shown by B. Harris and G. Segal that the composition \(Q_ 0(B\mu_ +)\to\text{BGL}(\mathbb{Z}[\zeta_ a])^ +\to\text{BGL}(\mathbb{F}_ q)^ +\) induces a surjection on homotopy groups with coefficients in \(\mathbb{Z}/\ell\). In fact, after localization at \(\ell\) the above map can be identified up to homotopy with the first projection \(r\) in a product decomposition. The aim of the present paper is to show that the map \(h: Q_ 0(B\mu_ +)\to\text{BGL}(\mathbb{Z}[\zeta_ a])^ +\) does not detect more in homotopy \(\text{mod }\ell\) than this surjection, by showing that after localization at \(\ell\) it is homotopic to \(h\circ s\circ r\), where \(s\) is a suitable right inverse to \(r\). This is proven by showing that for any finite \(\ell\)-group G and any map \(\text{BG}\to Q_ 0(B\mu_ +)\) the compositions with \(h\) and \(h\circ s\circ r\) are homotopic, using the generalized Burnside ring and representation rings.