I am interested in understanding (at least roughly, if no such a description exists) the group of automorphisms for the group $UT(n,p)$, of unitriangular matrices over the field $\mathbb{F}_p$ on $p$-elements. Unfortunately the online searching I've carried out wasn't quite helpful. If no description exists, maybe some information how the automorphisms behave on the set of subgroups of it may be helpful too.

For instance when $n=3$, I know we have that any $\phi \in Aut(UT(3,p))$, permutes the pairs of non-commuting elements of $UT(3,p)$ (which in that case happens always to be a generating set) and the subgroups of order $p^2$.

So am wondering if something similar exists (of course modified somehow) in the general case too.

If the above sounds quite general, the case where $n=4$, is of great importance for me either. Any comment might be useful, and of course references too.

Thanks!

"...and the automorphism group has respectively 19, 21 and 23 orbits on these subgroups.", you mean that the $Aut(UT(4,p))$-action on the set of subgroups of order $p^3$ yields these 19,21 and 23 orbits respectively or something different? Because it confuses me a little bit. $\endgroup$ – mayer_vietoris Sep 11 '17 at 13:30