Let $G$ be a group acting continuously on a free $\mathbb{Z}_p$-module of finite rank. Assume that $End_{G}(T)$ and $End_G(T/p)$ are the homotheties.
Is it possible that $End_{G}(T/p^n)$ contains more than the homotheties ?
When $T/p$ is absolutely irreducible, my guess would be that $T$ is also absolutely irreducible, and so is $T/p^n$ (I am aware that this heuristic is quite bad because what does it mean for a representation with coefficients in a ring to be irreducible when the ring is not simple ?)