Values at s=-1 of L-functions for relative quadratic extensions of number fields, and the Fitting ideal of the tame kernel

Fix a relative quadratic extension E/F of totally real number rields and let G denote the Galois group of order 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E and let O_E^S denote the ring of S_E-integers of E. Assume the truth of the 2-part of the Birch-Tate conjecture relating the order of the tame kernel K_2(O_E^S) to the value of the Dedekind zeta function of E at s=-1, and assume the same for F as well. We then prove that the Fitting ideal of K_2(O_E^S) as a Z[G]-module is equal to a generalized Stickelberger ideal. Equality after tensoring with Z[1/2][G] holds unconditionally.