Steinitz classes for Galois extensions of Dedekind rings

Galois extensions of commutative rings have been studied by Chase, Harrison, Rosenberg and others. Suppose B|A is such an extension with (finite) group G where both A and B are Dedekind rings. Then the Steinitz class (s) B|A is an element in the class group Cl(A) which vanishes if and only if B is a free A-module. It is shown that (s) B|A = 1 except possibly when the characteristic char(A) not equal 2 and G has a cyclic Sylow 2-subgroup not equal 1. In the exceptional case there is a unique (normal) subgroup H of G with index 2 and (s) B|A = (s) C|A where C = B (H) is the fixed ring. The remaining quadratic case is known and easily treated.