Elliptic curves and its Neron model

Let $$E$$ be an elliptic curve over $$\mathbb{Q}$$. For a prime $$p$$, let $$\mathcal{E}_p$$ denote its Neron model over $$\mathbb{Z}_p$$. Also, let $$\Phi_p(E)$$ denote the component group of $$\mathcal{E}_p$$.

The structure of $$\Phi_p(E)$$ is well-known, and I want to study it when $$E$$ has multiplicative reduction at $$p$$. First, if $$E$$ has split multiplicative reduction at $$p$$, then it is known that $$\Phi_p(E) \simeq \mathbb{Z}/{n\mathbb{Z}}$$, where $$n$$ is the (normalized) $$p$$-adic valuation of the discriminant of $$E$$. (I will admit this fact.) Next, if $$E$$ has non-split multiplicative reduction at $$p$$, then $$\Phi_p(E) \simeq \mathbb{Z}/{m\mathbb{Z}}$$, where $$m=1$$ or $$2$$ such that $$m \equiv n \pmod 2$$.

Here is my question: Suppose that $$E$$ has non-split multiplicative reduction at $$p$$. As above, we use the same notation ($$m$$, $$n$$, $$\Phi_p(E)$$ etc). We know the following.

1. There is a unramified quadratic extension $$L/{\mathbb{Q}_p}$$ such that $$E/{\mathcal{O}_L}$$ has split multiplicative reduction.

2. Neron model does not change under etale base change.

If the above two are correct, then the component group $$\Phi_p(E)$$ can be computed using the Neron model of $$E/L$$, where $$E$$ has split multiplicative reduction. Since the $$p$$-adic valuation does not change under the unramified extension, the component group of the Neron model of $$E/L$$ is isomorphic to $$\mathbb{Z}/{n\mathbb{Z}}$$. Thus, $$\Phi_p(E)$$ is also isomorphic to $$\mathbb{Z}/{n\mathbb{Z}}$$, which is not true if $$n>2$$.

I don't know where my argument fails. Please correct me!

The Galois group of $$L/{\mathbb Q_p}$$ acts on the component group over $$L$$ and the component group over $$\mathbb Q_p$$ should be the subgroup of elements fixed by the Galois group.