Let $F$ be a $p$-adic field with ring of integers $\mathfrak o$ and residue field $k$. Let $X$ be a smooth equidimensional projective scheme over $\mathrm{Spec}(\mathfrak o)$ of dimension $d$. Denote by $X_{\eta}$ the generic fiber and by $X_s$ the special fiber. We also fix a geometric generic point $\overline{\eta}$ giving rise to a geometric point $\overline s$ over $s$. Let also $\ell$ be a prime number different from $p$. We consider the étale cohomology of the varieties $X_{\eta}$ and $X_s$.
I think I have a confused understanding of how the nearby cycles sheaf works in this context.
Unless I am mistaken, one can build the nearby cycles as a derived functor $\mathrm R \Psi : D^+(X_{\eta},\overline{\mathbb Q_{\ell}}) \rightarrow D^+_G(X_{\overline s},\overline{\mathbb Q_{\ell}})$ where $D^+(\,\cdot\,,\overline{\mathbb Q_{\ell}})$ denotes the derived category of bounded above complexes of $\overline{\mathbb Q_{\ell}}$-étale sheaves on the given space, and the subscript $G$ means that it comes along with a continuous action of the absolute Galois group $G=\mathrm{Gal}(\overline{\eta}/\eta)$. This must first be defined over $\mathbb Z/\ell^{k}\mathbb Z$ and then extended to $\overline{\mathbb Q_{\ell}}$.
I'd like to understand the simplest example then, however I fail to find it (or I fail to understand it) in the litterature.
How should I think of $\mathrm R\Psi\overline{\mathbb Q_{\ell}}$ together with its Galois action ?
Because $X$ is smooth, I know that $\mathrm R^q\Psi\overline{\mathbb Q_{\ell}} = 0$ when $q\geq 1$ and $= \overline{\mathbb Q_{\ell}}$ when $q=0$. (Is this equality Galois equivariant, or is there some Tate twist to take the weights of the Frobenius into account ?)
And because $X$ is projective, I can write a Galois equivariant identity
$$\mathrm R\Gamma(X_{\overline{s}}, \mathrm R\Psi \overline{\mathbb Q_{\ell}}) \simeq \mathrm R\Gamma(X_{\overline{\eta}},\overline{\mathbb Q_{\ell}})$$
And thus, taking cohomology I should have $\mathrm H^i(X_{\overline{s}}, \mathrm R\Psi \overline{\mathbb Q_{\ell}}) \simeq \mathrm H^i(X_{\overline{\eta}},\overline{\mathbb Q_{\ell}})$. In both formula, when $\overline{\mathbb Q_{\ell}}$ is written alone, it should be understood as a complex having the single non-zero term $\overline{\mathbb Q_{\ell}}$ at position $0$.
Now, I would like to relate the left-hand side $\mathrm H^i(X_{\overline{s}}, \mathrm R\Psi \overline{\mathbb Q_{\ell}})$ with the classical etale cohomology of my variety $X_{\overline{s}}$. If I'm not mistaken, it is defined as the $i$-th cohomology group of some injective resolution of the complex $\mathrm R\Psi \overline{\mathbb Q_{\ell}}$, but this isn't a really concrete answer.
Nonetheless, the theory of perverse sheaves tell me that $\mathrm R\Psi$ preserves perversity, thus (?) $\mathrm R\Psi \overline{\mathbb Q_{\ell}}$ should be equal to the complex $\overline{\mathbb Q_{\ell}}$ shifted and twisted by some adequate integers, depending on $d$ and the dimension of the special fiber... ?
I'm not sure if talking about perverse sheaves is necessary to answer my problem.
In all the above, there must be some misconceptions or unprecise statements together with the questions. If you notice that something sounds wrong, please let me know in the comments.