Extensions of P-divisible Groups

Question

I'm having a lot of trouble understanding remark 1.7 on page 119 of Messing's "The Crystals Associated to Barsotti-Tate Groups".

In particular, I understand the argument given but I don't see where the conclusion "Therefore an extension of $G$ by $M$ admits no nontrivial automorphism and an extension is uniquely determined by its class in $Ext^1(G, M)$" comes from.

http://link.springer.com/book/10.1007%2FBFb0058301

Answer 1

Automorphisms of a given group extension $E$ of $G$ by $M$ form a torsor under $\operatorname{Hom}(G,M)$ (see SGA3 III 1.2.4; also this is very believable if one recalls the situation of ordinary groups). But $\operatorname{Hom}(G,M)$ is zero when $G$ is a BT-group over a scheme on which $p$ is nilpotent.