Let $k$ be a finite field of characteristic $p$. Let $H$ be a subgroup of $\rm{GL}_{n}(k)$ of order prime to $p$ where $n\geq2$. Assume that the representation $H\hookrightarrow \rm{GL}_{n}(k)$ is absolutely irreducible. Let $\text{ad}=M_{n}(k)$ with the action of $H$ coming from the adjoint action of $\rm{GL}_{n}(k)$ with the inclusion $H\hookrightarrow \rm{GL}_{n}(k)$. Let $\text{ad}^{0}$ be the matrices of trace zero in $ M_{n}(k)$, and we can view $\text{ad}^{0}$ as a $k[H]$-module. My question is the following:
For every irreducible $k[H]$-module $A\subset \text{ad}^{0}$, is there an element $x\in A$ and a nontrivial element $g\in H$ such that $x$ is $g$ invariant? (That is, the action of $H$ on $A$ is not free.)
One can show that the answer to the question is Yes when $n=2$ by checking all possible modules $ A$, cf. Lemma 4.1 in [Böckle, G., 1999. Explicit universal deformations of even Galois representations]. So, what if $n\geq 3$?
