induced map in homology on a fiber bundle

Let $$F \rightarrow E \rightarrow B$$ be a fiber bundle of compact connected smooth manifolds and $$B$$ simply connected. Suppose that there is a map $$f: E \rightarrow E$$ that covers a map $$g: B \rightarrow B$$. If the maps $$g_*: H_*(B, \mathbb{Q}) \rightarrow H_*(B, \mathbb{Q})$$ and $$(f_F)_*: H_*(F,\mathbb{Q}) \rightarrow H_*(F,\mathbb{Q})$$ are the identity map where $$f_F$$ denotes the restriction to the fiber, does it follows then that $$f_*$$ is the identity in the homology of $$E$$?

If I look at the Serre spectral sequence, this information tells me that $$f_*$$ is the identity just up to a filtration of $$H_*(E)$$, but it do not how to conclude that $$f_*$$ is the identity map or not