Firstly, let $k$ be an algebraically closed field of characteristic $p>0$.
Consider the local-local group scheme $$\alpha_p=\operatorname{Spec} k[\alpha]/(\alpha^p)\,,$$ then the claim in Oort's Which Abelian Surfaces are Products of Elliptic Curves is that $$\operatorname{End}_k(\alpha_p)\cong k\,.$$ My question is: why is this true? Surely $\alpha\mapsto\alpha^2$ is also a valid endomorphism; hence on top of the linear transformations, we must have that $\operatorname{End}_k(\alpha_p)$ is greater than $k$.
Background: I would like to count the number of supersingular principally polarised abelian surfaces and stumbled onto this question while reading Katsura and Oort's paper.