From the author’s introduction: Let \(K\) be a field and \(H\) a perfect finite group with \(H/Z(H) \cong A_ n\), the alternating group on \(n\) letters. In addition assume that \(| Z(H)| = 2\) if \(\text{char}(K) \neq 2\) and \(Z(H) = 1\) if \(\text{char}(K) = 2\). Let \(D\) be the set of elements of order 3 in \(H\) which correspond to the 3-cycles in \(A_ n\). The goal of this paper is to determine \(K[H]\)-modules \(W\) which have the property that elements in \(D\) have a quadratic minimal polynomial on \(W\), i.e. \(W(d^ 2 + d + 1) = 0\) for all \(d \in D\).