Let $A$, $B$ be abelian varieties over $\mathbb{Q}$, with corresponding Neron models $\mathcal{A}$, $\mathcal{B}$ over $X=Spec{\mathbb{Z}}$. Let $p$ be an odd prime of good reduction for both $A$ and $B$ such that we have an isomorphism $A[p]\cong B[p]$ over $\mathbb{Q}$. Keerthi Sampath has explained to me how this implies that we get an isomorphism $\mathcal{A}[p]\cong\mathcal{B}[p]$ of etale group schemes over  $U=Spec{\mathbb{Z}[\frac{1}{p}]}=X-Spec{\mathbb{F}_p}$. 

(a) I want to know if the same hypotheses as above also implies that we get an isomorphism $\mathcal{A}^0[p]\cong\mathcal{B}^0[p]$ on the identity components over $U$. 

(b) Is it true that $\mathcal{A}^0[p]$ (the $p$-torsion on $\mathcal{A}^0$) is the same as $\mathcal{A}[p]^0$ (the identity component of $\mathcal{A}[p]$)?