I am trying to understand some things related to elliptic curves and finite flat group schemes but I am a little bit confused.

Let $A$ be a supersingular elliptic curve over an algebraically closed field $K$ of characteristic $p$. Let $F: A \rightarrow A^{(p)}$ be the Frobenius isogeny. Then $\ker F$ is as a finite flat group scheme over $K$ isomorphic to $\alpha_p$, the kernel of the Frobenius on $\mathbb{G}_a$.

Now lets look at $F^2: A \rightarrow A^{(p^2)}$.

Is $\ker F^2: A \rightarrow A^{(p^2)}$ isomorphic (as a $K$-group scheme) to any one of the following group schemes:

$\ker F^2:\mathbb{G}_a \rightarrow \mathbb{G}_a$

$\ker F : W_2 \rightarrow W_2$ where $W_2$ is the group scheme of truncated Witt vectors

$A[p]$

$\alpha_p \oplus \alpha_p$

If $\ker F^2: A \rightarrow A^{(p^2)}$ is not isomorphic to one of them, I have an other question:

Is $\ker F^2: A \rightarrow A^{(p^2)}$ an extension of $\alpha_p$ over $\alpha_p$ and why not?

And a last one:

Is $\ker F^2: A \rightarrow A^{(p^2)}$ isomorphic to an other well-known group scheme?