Let $F\_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}\_4$, with Dynkin diagram
$$
\beta\_1-\beta\_2\Rightarrow\beta\_3-\beta\_4\,.
$$
Let $P\_{\{\beta\_2\}}$ be the minimal parabolic subgroup $\neq B$ such that its set of smiple roots is $\{\beta\_2\}$. Let
$$
\mu\,\colon F\_4\to\operatorname{Aut}(F\_4/P\_{\{\beta\_2\}})
$$
be the obvious morphism which sends an element $g\in F\_4$ to the translation $x\mapsto gx$. 



> 
> Is the morphism $\mu$ surjective? Or are there other automorphisms which do not come from the group? If there are more, how can they be described?
> 
> 
> 


This [question](https://mathoverflow.net/questions/74660/action-of-f-4-on-generalized-flag-manifolds) and the [reference](https://www.researchgate.net/publication/238258649/download) given in the comments there seem to be related. But I can't answer myself. 



Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$
The following paper says that your map $\mu$ is indeed surjective; the author attributes the result over $\mathbb C$ to Tits.



```
Demazure, M. Invent Math (1977) 39: 179. https://doi.org/10.1007/BF01390108

```

I also add the link 


[Automorphism group of flag manifolds?](https://mathoverflow.net/questions/160292/automorphism-group-of-flag-manifolds)


