Cohomology of pushforwards of vector bundles over the projectivization of a vector bundle

Question

Let $p:\mathcal E\to X$ be a vector bundle over a smooth projective variety $X$ and $p’:\mathbb P\mathcal E\to X$ its projectivization.

We consider, then, two vector bundles: $\mathcal B$ over the total space of $\mathcal E$, and $\mathcal B’$ over the total space of the projectivization $\mathbb P \mathcal E$.

EDIT: $\mathcal B’$ is the bundle obtained by $\mathcal B$ projectivizing the fibers of $\mathcal E$.

Is it possible to deduce the cohomology of $p_*\mathcal B$ by the cohomology of $p’_*\mathcal B’$ or vice versa?