Suppose $G$ is a profinite group, in fact in the applications I'm interested in it would be a $p$-adic analytic group similar to $GL_{n}(\mathbb{Z}_{p})$. Say $M$ is a profinite $G$-representation, that is, a limit of discrete, finite $G$-representations.
Let $(G_{i})_{i \geq 0}$ be a descending system of normal, open subgroups such that $\cap G_{i} = 0$. In the case of $GL_{n}(\mathbb{Z}_{p})$, we could take $G_{i}$ to be the kernels of the reduction maps $GL_{n}(\mathbb{Z}_{p}) \rightarrow GL_{n}(\mathbb{Z}/p^{i})$.
The question I'm interested in is as follows - what are the methods of computing the colimit of continuous cohomology groups
$\widetilde{H}^{k}(G, M) := \varinjlim_{i} H^{k}_{c}(G_{i}, M)$?
In particular, can it be computed through some Lie-theoretic methods?
Note that a profinite $G$-module is not necessarily admissible, that is, it is not necessarily the case that every $m \in M$ is stabilized by an open subgroup of $G$. Instead, the colimit $\varinjlim_{i} H^{0}_{c}(G_{i}, M)$ is the largest admissable subrepresentation of $M$. Thus, perhaps in an appropriate category of modules one could recognize $\widetilde{H}^{k}(G, -)$ as the derived functors of "largest admissible subrepresentation".
A worked-out example. Say $G = \mathbb{Z}_{p}$, in this case we can take $G_{i} = p^{i} \mathbb{Z}_{p}$. Let $M = \mathbb{Z}_{p}$ with the trivial action, in this case the usual continuous cohomology groups are given by
$H^{0}(\mathbb{Z}_{p}, \mathbb{Z}_{p}) \simeq \mathbb{Z}_{p}$, $H^{1}(\mathbb{Z}_{p}, \mathbb{Z}_{p}) \simeq \mathbb{Z}_{p}$,
while the higher ones vanish.
Since we equipped $M := \mathbb{Z}_{p}$ with the trivial action, this is an admissable module and we have $\widetilde{H}^{0}(\mathbb{Z}_{p}, \mathbb{Z}_{p}) \simeq \mathbb{Z}_{p}$. On the other hand, the situation with $k = 1$ is more interesting. The topological generator of $H_{c}^{1}(\mathbb{Z}_{p}, \mathbb{Z}_{p})$ is given by the identity homomorphism and we deduce that the maps in the defining colimit
$\widetilde{H}^{1}(\mathbb{Z}_{p}, \mathbb{Z}_{p}) := \varinjlim_{i} (H^{1}_{c}(\mathbb{Z}_{p}, \mathbb{Z}_{p}) \rightarrow H^{1}_{c}(p\mathbb{Z}_{p}, \mathbb{Z}_{p}) \rightarrow \ldots)$
corresponds to multiplication by $p$, yielding $\widetilde{H}^{1}(\mathbb{Z}_{p}, \mathbb{Z}_{p}) \simeq \mathbb{Q}_{p}$. I'm interested in finding methods that predict this kind of answer, and that would make it possible to compute such colimits in cases where the actual continuous cohomology is unknown.
(Note that in the example I gave, it's not hard to see that $\widetilde{H}^{k}(\mathbb{Z}_{p}, \mathbb{Z}_{p})$ is necessarily rational for $k > 0$, and then it can be computed through the rational cohomology formula due to Lazard. However, the cases I'm interested would be rather asking about cohomology with coefficients in a local field of characteristic $p$.)
