ag.algebraic geometry - Abelian varieties over local fields

Let $K$ be a local field of characteristic zero, $k$ its residue field, $R$ its ring of integers and $p$ the characteristic of the residue field $k$. Let $G$ be the Galois group of $K$, $Isubset G$ the inertia group and $P$ the maximal pro-$p$ subgroup of $I$. Let $I_t:=I/P$.

Let $A_0$ be an abelian scheme over $R$ with generic fibre $A$. Then $A[p]$ is an $I$-module.
Let $V$ be a Jordan-Hölder quotient of the $I$-module $A[p]$.
I am interested in the representation $Ito Aut(V)$.

Question (*): Is it true that $P$ acts trivially on $V$?

(I have seen that there are results
of Raynaud and Serre on the "action of $I_t$ on $V$". I want to study these things, but I am already stuck
with Question (*) at the moment, i.e. with the question whether $I_t$ acts at all.)

Maybe someone can help?