Ideal class groups of CM-fields with non-cyclic galois action

Suppose that L/k is a finite and abelian extension such that k is a totally real base field and L is a CM-field. We regard the ideal class group Cl _L, of L as a Gal(L/k)-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for L/k times the annihilator ideal of the roots of unity in L is in the Fitting ideal of Cl_L, and also a problem whether it is in the Fitting ideal of the Pontrjagin dual (Cl_L)^v. We systematically construct extensions L/k for which these properties do not hold, and also give numerical examples.