Fitting ideals of class groups for CM abelian extensions

Let $K$ be a finite abelian CM-extension of a totally real field $k$ and $T$ a suitable finite set of finite primes of $k$. We determine the Fitting ideal of the minus component of the $T$-ray class group of $K$, except for the $2$-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal.