Let $R$ be a commutative ring, $M$ is a finitely presented module: $$ R^m \xrightarrow{A} R^n \to M \to 0. $$ Fitting ideal $\mathrm{Fit}_k(M)$ of $M$ is defined as ideals generated by all $(n-k) \times (n-k)$ minors of $A$. It is independent of the free presentation.
How Fitting ideals are related to annihilators of exterior powers of $M$? A classical result says $$ \mathrm{Ann}(M)^l \subseteq \mathrm{Fitt}_0 (M) \subseteq \mathrm{Ann}(M), $$ where $l$ is the minimal number of generators of $M$. Are there relations between other Fitting ideals and $\mathrm{Ann}(\wedge^p M)$ (perhaps, with additional assumptions on R)?