Is $k(G/S_p)$ a semi-simple $k(G)$-module?

Let $G$ be a finite group, let $k$ be a field of characteristic $p$, and let $S_p$ be a Sylow $p$-subgroup of $G$. It is a well-known fact from modular representation theory that every irreducible representation factors through any normal $p$-subgroup of $G$. In particular, it factors through the maximal normal $p$-subgroup, i.e., $N = \cap_{g \in G}gS_pg^{-1}$. Also, every semisimple module factors through $N$.

I want to prove this is the best that can be done, i.e., there exists a semisimple module whose kernel is exactly $N$. Here $V = k(G/S_p)$ satisfies that the kernel is exactly $N$, but I am unable to prove that it is semisimple.

Any suggestion on how to prove the semisimplicity of $V$ (if it is semisimple) or any alternate methods to prove the original question would be greatly appreciated.
Thank you.