I start with a hopelessly broad question: what is known about the structure of the automorphism group of a (smooth, connected) unipotent group (over a field), and particularly about the structure of diagonalisable subgroup schemes of the automorphism group? It would be nice, but is not essential, if I could assume a non-algebraically closed field of definition.

I have two, I think sufficiently specifically focussed, questions. Suppose that $\Gamma$ is a diagonalisable group scheme acting on a unipotent group $U$.

Can we extend the action of $\Gamma$ to some torus? (That is, are there a torus $T$ acting on $U$ and a map $\Gamma \to T$ such that the obvious diagram commutes?)

Can we extend the action of $\Gamma$ to some diagonalisable group scheme $\tilde\Gamma$ with the property that there is some cocharacter $\tilde\lambda$ of $\tilde\Gamma$ such that $\langle\tilde\chi, \tilde\lambda\rangle > 0$ for all non-0 weights $\tilde\chi$ of $\smash{\tilde\Gamma}^\circ$ on $\mathrm{Lie}(U)$? (If necessary, I can assume here that $\Gamma$ is a torus.)

EDIT: nfdc23 points out that this is nearly a question about split unipotent groups, since wound groups admit no non-trivial torus actions. In fact, this is not far off my original motivation. In §15.13(b) of his famous book, Borel mentions a 1968 article (MR) with Springer in which he investigates rationality questions for algebraic groups. Particularly, §9 of that paper says quite a bit about the structure of unipotent groups equipped (most of the time) with a fixed-point-free torus action. (The result quoted by Borel says, in particular, that they must be split.) In most cases, it seems that the proofs work just as well if we have instead just an action by a diagonalisable group scheme. I say this on the basis just of working line-by-line through the proofs, but a positive answer to (1) would save such (as someone, I think Bushnell and Kutzko, memorably put it in another context) "lame and automatically suspect" reasoning.

In the course of these investigations, I realised that I'd been behaving as if arbitrary unipotent groups behaved like unipotent radicals of parabolic subgroups, in the sense at least that they were of the form $U(\lambda)$ (cf. Springer, §3.2.15). It's easy to see that this isn't true, but I was wondering how badly it could fail. That's the motivation for (2).