  In this paper, for a CM abelian extension $K/k$ of number fields, we propose
a conjecture which describes completely the Fitting ideal of the minus part of
the Pontryagin dual of the $T$-ray class group of $K$ for a set $T$ of primes
as a ${\rm Gal}(K/k)$-module. Here, we emphasize that we consider the full
class group, and do not throw away the ramifying primes (namely, the object we
study is not the quotient of the class group by the subgroup generated by the
classes of ramifying primes). We prove that our conjecture is a consequence of
the equivariant Tamagawa number conjecture, and also prove that the Iwasawa
theoretic version of our conjecture holds true under the assumption $\mu=0$
without assuming eTNC.
