Maximal unramified quotient of $E[p]$ for the action of $G_{\mathbb{Q}_p}$

Let $ E$ be an elliptic curve with good and ordinary reduction at an odd prime $ p$ . Suppose $ E[p]$ denotes the $ p$ -torsion points of $ E$ and $ G_{\mathbb{Q}_p} := \text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ .

In the article `Selmer group and congruences (page 6)’, Greenberg says that one can characterize $ \widetilde{E}[p]$ as the maximal unramified quotient of $ E[p]$ for the action of $ G_{\mathbb{Q}_p}$ where $ \widetilde{E}$ denotes the reduction of $ E$ in $ \mathbb{F}_p$ .

This is so because $ p$ is assumed to be odd and therefore the action of the inertia subgroup of $ G_{\mathbb{Q}_p}$ on the kernel of the reduction map $ \pi: E[p] \longrightarrow \widetilde{E}[p]$ is nontrivial.

It will be every helpful if someone can explain how `$ p$ being odd’ is playing a role in proving the non trivial action of the inertia subgroup on the kernel of the reduction map $ \pi$ ?