Let \(G\) be a finite group and let \(X\) be a finite pointed \(G\)-CW complex with a free \(G\)-action away from the base point. By the result of J. F. Adams [Lect. Notes Math. 1051, 483–532 (1984; Zbl 0553.55010)] we know that for any finite pointed CW complex \(Y\) there is an isomorphism \([\Sigma^\infty Y, \Sigma^\infty (X/G)]\cong [\Sigma^\infty Y, \Sigma^\infty X]^G\). This means that in the stable homotopy category there is an isomorphism between the derived \(G\)-orbits and the derived \(G\)-fixed points of \(\Sigma^\infty X\). In fact, according to L. G. Lewis jun. et al. [Equivariant stable homotopy theory. With contributions by J. E. McClure. Berlin etc.: Springer-Verlag (1986; Zbl 0611.55001)], this holds for general spectra. The purpose of this paper is to realize this isomorphism with the spectrum level construction in the category of orthogonal spectra. Let \(N\) be a normal subgroup of \(G\) and let \(\mathcal{U}\) denote a complete \(G\)-universe. Then the main theorem is that for any orthogonal \(G\)-spectrum \(\mathbf X\) there is an equivariant map of orthogonal \(G/N\)-spectra \[ A : E\mathcal{F}(N)_+\wedge_N\mathbf X\to Q^\mathcal{U}(\mathbf X)^N \] such that it is natural in \(\mathbf X\) and becomes a stable weak equivalence when \(\mathbf X\) is cofibrant and \(N\)-free. Here \(\mathcal{F}(N)\) denotes the family of subgroups \(H\) of \(G\) such that \(H\cap N=1\) and \(E\mathcal{F}(N)\) the universal space for \(\mathcal{F}(N)\). From this, the original Adams isomorphism mentioned above can be deduced as follows. By using the results of M. A. Mandell and J. P. May [Mem. Am. Math. Soc. 755, 108 p. (2002; Zbl 1025.55002)] we first have a stable weak equivalence \(E\mathcal{F}(N)_+\wedge_N\mathbf X\simeq \mathbf X/N\) of orthogonal \(G/N\)-spectra which is induced by the projection \(E\mathcal{F}(N)_+\wedge\mathbf X\to\mathbf X\). Hence from the main theorem it follows that \(\mathbf X/N\simeq Q^\mathcal{U}(\mathbf X)^N\), so we have \[ [\mathbf Y,\mathbf X/N]^{G/N} \cong [p^*\mathbf Y,\mathbf X]^G \] for any cofibrant \(G/N\)-spectrum \(\mathbf Y\) where \(p : G\to G/N\) is the projection. Here, take \(\mathbf X=\Sigma^\infty X\), \(\mathbf Y=\Sigma^\infty Y\) and put \(N=G\), then the desired isomorphism follows from this.