Let $p$ be a prime, $G$ be a finite group of order $p^a$. Let $M$ be a $\mathbb{Z}[G]$-module. Then $H^n(G, M)$ is annihilated by $p^a$ for all $n \geq 1$ (see e.g. Brown, Corollary III.10.2).

In particular this is true for $\mathbb{Z}_p[G]$-modules, and I was wondering if this version of the result can be generalized to local fields other than $\mathbb{Q}_p$. More precisely, let $\mathbb{K}$ be a non-Archimedean local field with ring of integers $\mathfrak{o}$, uniformizer $\omega$, maximal ideal $\mathfrak{p}$, residue field $\mathfrak{k}$ of characteristic $p$. Let $G$ be a finite $p$-group and $M$ an $\mathfrak{o}[G]$-module. What is the least $a$ such that $H^n(G, M)$ is annihilated by $\omega^a$?

It should be easy to use the statement at the beginning of this post to generalize this to any local field of characteristic $0$, and $a$ should still be uniformly bounded, with the bound depending only on the order of $\mathfrak{k}$ and the order of $G$. But what about characteristic $p$? This case seems to be substantially different.

I think this question is interesting in general, but there is probably no chance to get a general answer without assuming anything extra. So let me point out the specific setting that I am interested in:

$n = 1$ and especially $n = 2$: this comes up in an extension problem so I only really care about these degrees.

$M$ is a free $\mathfrak{o}/\mathfrak{p}^k$-module. So I already know that $H^n(G, M)$ is annihilated by $\omega^k$, that is $a \leq k$. But I would like to get better estimates, ideally I would like to find a bound of the form $a \leq a(k)$ such that $(k - a(k)) \to \infty$ as $k \to \infty$. In characteristic $0$ I can get $a(k)$ to be a constant so this is a significant weakening, but still somewhat strong.

Even though I am interested in the general case, I have trouble understanding even the first example: $G = \mathbb{Z}/2\mathbb{Z}$ and $\mathbb{K} = \mathbb{F}_2((X))$, so $\omega = X$ and $\mathfrak{o} = \mathbb{F}_2[[X]]$. Explicitly: if $M$ is a free $\mathbb{F}_2[X]/X^k$-module with an action of $G$, what is the least $a$ such that $X^a H^n(G, M) = 0$ for $n = 1, 2$? After trying a bit I think that one cannot do better than $a(k) = k/2$ (see the previous point), but I am not able to prove that this works.