This seems like a really basic question, but I somehow don't know and haven't been able to find the answer.

I suspect that (at least under suitable assumptions) there should be a relation between the following two constructions, but I'm looking for a precise theorem.

If I have a family $X$ over a disk, which I'm thinking of as a degeneration of smooth fibers $X_t$ to a singular fiber $X_0$, then there are two natural perverse sheaves I can cook up on $X_0$: one is the intersection cohomology of $X_0$, and the other is the nearby cycles of the intersection cohomology (which is constant) on $X-X_0$. What is the relation between these two?

To make things a little more precise: let $\eta \rightarrow S \leftarrow s$ be a henselian trait, $X \rightarrow S$ a map with $X_s$ and $X_{\eta}$ the special and generic fibers. I'm imagining that $X_{\eta} \rightarrow \eta$ is smooth, but perhaps this is unnecessary. How can I describe $IC(X_s, \mathbb{Q}_{\ell}))$? in terms of $\Psi(IC(X_{\eta}, \mathbb{Q}_{\ell}))$? (I expect an answer to involve the weight filtration and monodromy operator on nearby cycles, in addition to the "bare sheaf".)