Let $R$ be a complete local $\mathbf{Z}_p$-algebra, for some prime $p$. In the 1970 paper Group schemes of prime order by Oort and Tate, they write down an explicit finite flat group scheme $G_R(a, b)$ of rank $p$, for each pair of elements $a,b \in R$ satisfying $ab = p$, and show that every FFGS over $R$ of rank $p$ is isomorphic to one of these.

What are the homomorphisms (of $R$-group schemes) from $G_R(a, b)$ to $G_R(a', b')$?

The underlying scheme of $G_R(a, b)$ is $\{ X : X^p - aX = 0\}$. If we look for homomorphisms which are "linear", $X \mapsto \lambda X$ for some $\lambda$, then (after a bit of unravelling) we conclude that $\lambda$ has to be a point of the $R$-scheme $$\{ Z : Z(aZ^{p-1} - a') = Z(b - b'Z^{p-1}) = 0\}.$$ (This is a group scheme, but neither finite nor flat over $R$ in general, although it is a FFGS if $\{a, b'\}$ generate the unit ideal of $R$.)

**Are these all the homomorphisms** $G_R(a, b) \to G_R(a', b')$?

(The answer is clearly "yes" for $p = 2$. It is also "yes" for $p = 3$ via a deeply nasty polynomial computation.)