Let G be a finite group, N a normal subgroup of G and k a field. Let \(\tilde G=G/N\). The authors prove that there is a kG-module M such that, whenever L is a \(k\tilde G\)-module which has projective covers Q and \(\tilde Q\) as a kG-module and as a \(k\tilde G\)-module respectively, then Q and \(\tilde Q\otimes M\) have the same composition factors. Moreover, let P be a projective kG-module and \(\tilde P\) be a projective \(k\tilde G\)-module. Then \(P/Rad(P)\) and \(\tilde P/Rad(\tilde P)\) are isomorphic if P and \(\tilde P\otimes M\) have the same composition factors. This generalizes results due to \textit{J. L. Alperin}, \textit{M. J. Collins} and \textit{D. A. Sibley} [Bull. Lond. Math. Soc. 16, 416-420 (1984; Zbl 0533.20003)].