Let E signify a totally real Abelian number field with a prime power conductor and ring of p-integers R_E for a prime p. Let G denote the Galois group of E over the rationals, and let chi be a p-adic character of G of order prime to p. The odd-primary results in this paper depend on the Bloch-Kato conjecture, while the two-primary results are non-conjectural. Theorem A calculates, under a minor restriction on chi, the Fitting ideals of K_n(R_E;Z_p)(chi) over Z_p[G](chi). Here we require that n = 2 mod(4). These Fitting ideals are principal, and generated by a Stickelberger element. This gives a partial verification and also a strong indication of the Coates-Sinnott conjecture. We also discuss (co)-descent for higher K-groups, and prove in Theorem B a Hilbert Theorem 90 type of result for the transfer map in higher K-theory of number fields.