Recall that for \(A\) a unital, monotone complete \(C^*\)-algebra, \(G_ 1\) and \(G_ 2\) countably infinite groups, and \(\alpha_ 1\), \(\alpha_ 2\) actions of \(G_ 1\) and \(G_ 2\), respectively, as \(*\)-automorphisms of \(A\), the actions \(\alpha_ 1\) and \(\alpha_ 2\) are called weakly equivalent if there exists an isomorphism of the monotone cross-products \(M(A, G_ 1, \alpha_ 1)\) and \(M(A, G_ 2, \alpha_ 2)\) which maps the canonical image of \(A\) in the former onto the canonical image of \(A\) in the latter. The main result obtained is the following: ``Let \(A\) be a monotone complete factor. Let \(G_ 1\) and \(G_ 2\) be two discrete groups. Let \(\alpha_ 1\) and \(\alpha_ 2\) be free actions on \(A\) of, respectively, \(G_ 1\) and \(G_ 2\). If \((A, G_ 1, \alpha_ 1)\) and \((A, G_ 2, \alpha_ 2)\) are weakly equivalent, then \(G_ 1\) and \(G_ 2\) are isomorphic.'' This is in sharp contrast with the Sullivan-Weiss-Wright theorem which asserts that for \(A\) the Dixmier algebra, \((A, G_ 1, \alpha_ 1)\) and \((A, G_ 2, \alpha_ 2)\) are always weakly equivalent for any countably infinite discrete groups \(G_ 1\) and \(G_ 2\). An example of a setting in which the present result applies is when \(A= \widehat F\), the regular \(\sigma\)-completion of the Fermion algebra \(F\) [see the second author, J. Lond. Math. Soc., II. Ser. 12, 299-309 (1976; Zbl 0316.46055)].