Commutative quartic P-Galois extensions over a field of characteristic 2

Let A/R be a ring extension and P a subset of Hom(A(R),A(R)). In his paper [5], K. Kishimoto introduced the notion of a P-Galois extension and gave several basic properties of these extensions. The author showed that these extensions are closely related to Hopf Galois extensions and the structure of quadratic or cubic P-Galois extensions over a field were given in [9] and [10]. Recently,the author classify commutative quartic P-Galois extensions over a field of characteristic not 2 in [11]. Continuing [11], we treat commutative quartic P-Galois extensions over a field of characteristic 2.