# quasi-finite group schemes and associated Galois modules

Let $p$ be an odd prime.

Let $A$ be an abelian variety over $\mathbb{Q}$ and suppose that it has semistable reduction at $p$. Let $\mathcal{A}$ denote the Neron model of $A$ over $\mathbb{Z}_p$. Now consider a group scheme $A[p]$, the kernel of multiplication by $p$ in $A$. Let $\mathcal{A}[p]$ denote the scheme-theoretic kernel of multiplication by $p$ in $\mathcal{A}$, which is a quasi-finite flat separated group scheme of $p$-power order. Let $G$ be a subgroup scheme of $A[p]$ and $\mathcal{G}$ its Zariski closure in $\mathcal{A}$, which is a closed subgroup scheme of $\mathcal{A}[p]$. As $\mathcal{G}$ is a quasi-finite flat separated group scheme over $\mathbb{Z}_p$, by the structure theorem, there is an exact sequence $$0 \to FG \to \mathcal{G} \to EG \to 0$$ where $FG$ is a finite flat group scheme over $\mathbb{Z}_p$ and $EG$ is an etale quasi-finite group scheme whose closed fiber is trivial (Lemma 1.1 of Mazur's paper, Rational isogenies of prime degree, https://link.springer.com/content/pdf/10.1007/BF01390348.pdf).

Let $V:=G(\overline{\mathbb{Q}})$ denote the Galois module associated to $G$. Suppose that all the Jordan-Holder constituents of $V$ are either $\mathbb{Z}/p\mathbb{Z}$ or $\mu_p$. I have two questions:

Question 1: In this situation, what could $EG$ be? Is always $\mathcal{G}$ finite? (Namely, is $EG$ always trivial?)

Question 2: Let $a$ be the number of copies of $\mathbb{Z}/p\mathbb{Z}$ in the Jordan-Holder constituents of $V$. Let $b$ be the number of points of $\mathcal{G}$ in $\overline{\mathbb{F}}$, i.e., $b=\#\mathcal{G}(\overline{\mathbb{F}})$.

Then, what's the relation between $a$ and $b$? Is $b$ always equal to $p^a$?