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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 ?)

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First a general remark: there is an article of Carayol in the $p$-adic monodromy volume (edited by Mazur and Stevens) which deals with the issue of passing information from $T/p$ to $T$ (and in particular with the fact that it doesn't make rigorous sense to say that $T$ is abs. irred.). You could also look at Mazur's original article on deformations of Galois representations.


Now for your question: Let $\varphi: T/p^n \to T/p^n$. Reducing mod $p$, $\varphi$ becomes a scalar. Let $a \in \mathbb Z_p$ be a lift of this scalar. Then $\varphi -a$ acts trivially on $T/p$, and so maps $T/p^n$ to $pT/p^nT \cong T/p^{n-1}$. Thus $\varphi-a$ may be obtained as $p$ times an endomorphism of $T/p^{n-1}T$. Arguing by induction on $n$, we find that $\varphi - a$ is scalar, and hence so is $\varphi$ itself.

Thus your question has a positive answer (and in fact it is enough to assume that $T/pT$ has only scalar endomorphisms). Also, one can replace $\mathbb Z_p$ by any complete local Noetherian $\mathbb Z_p$-algebra (the same argument will work, and it is related to the fact that if $T/pT$ has only scalar endomorphisms then it has a representable deformation functor).