Let k be a field of characteristic 2. The representation theory of the alternating group A4 over k has an essentially twofold character depending on whether k does or does not contain the cubic root of unity. The case when x2 + x + 1 is reducible over k is reflected in the case when k is algebraically closed, which has been treated in detail by several authors (see S. B. Conion [Z], E. Kern [9]). The case when x*+x + 1 is irreducible over k has been considered explicitly only recently (see U. Schoenwaelder [lo]). Of course, there is a well-known general procedure (see D. G. Higman [8]) to find the represenations of A, in both cases. This is due to the fact that A4 contains the Klein 4-group V, as a normal subgroup and the representations of V, are well-understood (see V. A. Bagev [ 11, A. Heller, I. Reiner [7)). It is this procedure that has been used by U. Schoenwaelder in the case where x2 + x + 1 is irreducible over k. An inherent diffl~ulty of this method appears in the explicit listing and full understanding of certain representations, viz. those belonging to the one-parameter family of kA,-modules which is induced from the one-parameter family of indecomposable kV,-modules. Whereas in the case of V, these representations are indexed by the irreducible polynomials over k, the corresponding index set appears to be much more involved in the case A,: One has to consider an action of the cyclic group of order 3 on the polynomials and determine the corresponding orbits. However, there is an alternative approach presented in our paper, which shows that also in the case of AA, the one-parameter family of representations may be indexed again by the set of all irreducible