Fitting ideals of Class groups in Carlitz-Hayes cyclotomic extensions

We generalize some results of Greither and Popescu to a geometric Galois cover $X\to Y$ which appears naturally for example in extensions generated by ${\mathfrak{p}}^n$-torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz-Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in \cite{ABBL}) with Goss $\zeta$-function.