Let $F\rightarrow E\rightarrow B$ be a fiber bundle. Let $\pi_1$ be the fundamental group of $B$ with base point say $b_0$. In the following we are considering cohomology with coefficients in $\mathbb{Z}$ (but I would not mind to consider coefficients in $\mathbb{Q}$ if torsion elements turn out to be troublesome regarding the question). Each (equivalence class) of loop $[\gamma]\in \pi_1$ induces an automorphism on $F$ which provides an action of $\pi_1$ on $H^*(F)$.

Not being an algebraic topologist, my naive intuition tells me that it should sometimes happen that \begin{align} (\star) \hspace{1cm} H^*(E)\simeq H^*(B)\otimes H^*(F)^{\pi_1} \end{align} At least we know that this is indeed the case when $\pi_1$ acts trivially on $H^*(F)$ and that the spectral sequence degenerates at the second page (i.e. all differentials $d^k$ are trivial for $k\geq 2$). If I were fluent regarding spectral sequences I would not be asking these two questions but since I'm not here they are:

**Q1:** Do we know other (general) situations for which $(\star)$ is known to be an isomorphism (I don't really care if it is canonical or not) ?

Analogues of this question could as well be asked in the setting of discrete groups. For example say that $1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1$ is a **split** short exact sequence of discrete groups.

**Q2** Are there (general) situations for which $H^*(E)\simeq H^*(G)\otimes H^*(A)^G$ ?

**added:** Here is one interesting example (if my calculations are correct) where ($\star$) is an isomorphism. Let $S^1$ be the circle and set $B=(S^1)^{b}$ and $F=(S^1)^a$. Consider a group homomorphism $\rho:\pi_1=\pi_1(B)\simeq \mathbb{Z}^b\rightarrow GL_a(\mathbb{Z})\simeq Aut(F)$. Then it seems to me that ($\star$) holds true. Let me be more explicit about an example coming from number theory. Take for example $L=\mathbb{Z}[\sqrt{2}]$ and $\Lambda=\epsilon^{\mathbb{Z}}$ where $\epsilon=3+2\sqrt{2}$. Then $\Lambda$ acts naturally on $L$ and one may consider the arithmetic group $\Gamma=L\rtimes \Lambda$. Let $\mathfrak{h}$ be the Poincare upper half-plane. Then an element of the group $(v,\lambda)\in\Gamma$ acts naturally on $\mathfrak{h}^2$ by the usual rule
\begin{align*}
(v,\lambda)*(z_1,z_2)=(\lambda^{(1)}z_1+v^{(1)},\lambda^{(2)}z_2+v^{(2)})
\end{align*}
This gives rise to the following bundle
\begin{align*}
F=\mathfrak{h}^2/V\rightarrow E=\mathfrak{h}^2/\Gamma\rightarrow B=\mathbb{R}_{>0}^2/\Lambda
\end{align*}
The first term is homotopy equivalent to $(S^1)^2$ and the second to $S^1$. Now if one considers the de Rham cohomology group $H^*(E)$ then it seems to me that all the closed differential forms on $E$ which are $\Gamma$-invariant are obtained from the following tensor product:
\begin{align}
\{\mathbb{R}1+\mathbb{R}dx_1\wedge dx_2 \} \otimes \{\mathbb{R}1+\mathbb{R}\frac{dy_1}{y_1}\},
\end{align}
where $z_j=x_j+i y_j\in \mathfrak{h}$ for $j=1,2$. Note that $\frac{dy_1}{y_1}$ represents the same cohomology class as $-\frac{dy_2}{y_2}$ in $H^*(B)$ since the function $\log y_1y_2$ is $\Gamma$-invariant.