I am reading through Weibel's chapter on Galois Cohomology and there he defines profinite group cohomology as the right derived functors of the $G$-invariants functor, but restricting to $C_G$, a subcategory of $G$-modules consisting of all discrete modules (i.e. modules with the discrete topology and continuous $G$-action). So essentially we are deriving the composition of the inclusion of $C_G$ into $G$-mod and the invariants functor $G$-mod to $Ab$.
Is there a nice way to describe what happens to derived functors when we precompose with an inclusion like this? In more precise terms, if we have an abelian category with enough injectives $\mathscr{A}$, a left exact functor $F: \mathscr{A}\rightarrow \mathscr{C}$ and a subcategory $\mathscr{B}$ of $\mathscr{A}$ which is also abelian and has enough injectives, what can we say about the right derived functors of the restriction $F:\mathscr{B}\rightarrow \mathscr{C}$? I have seen that the Grothendieck spectral sequence gives information about the derived functors of a composition of functors, but I do not understand spectral sequences well enough to tell whether it can be applied here.
Note: Weibel proves there are enough injectives in $C_G$ by showing the inclusion functor $C_G\rightarrow G-mod$ has a right adjoint, namely the discretization functor, which takes a $G$-module $A$ and outputs the union of all $A^U$, where $U$ ranges over all onen subgroups of the profinite group $G$. (see here: http://math.stanford.edu/~conrad/210BPage/handouts/profcohom.pdf). Since the discretisation functor is right adjoint to the exact inclusion functor, it preserves injectives.