Let $X$ be a smooth complex alegebraic variety and $f: X \to \mathbb{C}$ a proper morphism that is smooth away from $0 \in \mathbb{C}$. Let $0 \neq b \in \mathbb{C}$. The pullback map $H^n(X, \mathbb{Q}) \to H^n(X_b, \mathbb{Q})$ factors through the $\mathbb{Z}$-invariant subspace $H^n(X_b, \mathbb{Q})^{\mathbb{Z}} \subseteq H^n(X_b, \mathbb{Q})$ where $\mathbb{Z}$ (considered as the fundamental group of $\mathbb{C}^*$) acts by monodromy on cohomology of the family $X \times_{\mathbb{C}} \mathbb{C} ^* \to \mathbb{C}^*$. The "local invariant cycle theorem" says that $H^n(X, \mathbb{Q}) \to H^n(X_b, \mathbb{Q})^{\mathbb{Z}}$ is surjective.
My question: what can we say about the image of $H^n(X, \mathbb{Z}) \to H^n(X_b, \mathbb{Z})^{\mathbb{Z}}$?
In my application I know that the monodromy has order $l$. Let $H^n(X, \mathbb{Z})_{Fr}$ be the free part. I'm happy if I can bound the index of $\left(\mathrm{Im}\, H^n(X, \mathbb{Z}) \cap H^n(X_b, \mathbb{Z})_{Fr} \right) \subseteq H^n(X_b, \mathbb{Z})^{\mathbb{Z}}_{Fr}$