This paper considers various kinds of multiplicative transfers, namely Evens transfer, Kozlowski transfer and normalized Evens transfer in singular and Galois cohomologies, which have been introduced as tools for solving the problem of determining characteristic classes of direct image bundles for covering maps (Fulton–MacPherson) or Stiefel–Whitney classes of induced Galois representation for finite separable extension of commutative fields of characteristic not 2 (Kahn).

We show that normalized Evens transfer is covariant, when restricted to homogeneous elements or Stiefel–Whitney classes, in the case of singular cohomology. We deduce that, in the case of Galois cohomology, Kozlowski transfer and the transfer involved by Kahn coïncide on invertible elements. We conclude that the Kahn formula can be viewed as an inflation of the Fulton–MacPherson formula.