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! 



Automorphism group of $UT(n,p)$, the group of unitriangular matrices over the field $\mathbb{F}_p$
* for size = 3
<https://groupprops.subwiki.org/wiki/Unitriangular_matrix_group:UT(3,p)#Automorphisms>
Automorphisms
The automorphisms essentially permute the subgroups of order  

containing the center, while leaving the center itself unmoved.
* over $F\_2$, any size: [Maginnis J. S., (1993/11)."Outer Automorphisms of Upper Triangular Matrices." Journal of Algebra 161(2): 267-270.](https://doi.org/10.1006/jabr.1993.1218)
Abstract: The outer automorphism group of the upper triangular matrices over the field of two elements is calculated. A. J. Weir (Proc. Amer. Math. Soc.6 (1955), 454-464) performed a similar calculation for Fields of odd characteristic, and we borrow the term extremal auto ....
* Any field(may be even ring) [International Journal of Algebra, Vol. 7, 2013, no. 15, 723 - 733
HIKARI Ltd,
The Automorphism Group of the Group of
Unitriangular Matrices over a Field1
Ayan Mahalanobis](http://dx.doi.org/10.12988/ija.2013.3991)
Abstract. This paper finds a set of generators for the automorphism group
of the group of unitriangular matrices over a field. Most of this paper is an
exposition of the work of V.M. Lev˘chuk, part of which is in Russian. Some
proofs are of my own.
From the paper: 
The automorphism group of the group of unitriangular matrices over a field
was studied by many [2–4]. In this direction, the first paper was in Russian,
published by Pavlov in 1953. Pavlov studies the automorphism group of unitriangular
matrices over a finite field of odd prime order. Weir [4] describes
the automorphism group of the group of unitriangular matrices over a finite
field of odd characteristic. Maginnis [3](https://mathoverflow.net/q/106521/10446) describes it for the field of two elements
and finally Lev˘chuk [2](http://dx.doi.org/10.12988/ija.2013.3991) describes the automorphism group of the group
of unitriangular matrices over an arbitrary ring.
In this expository article, we shall study the automorphism group of the
group of unitriangular matrices over an arbitrary field F. There are two most
commonly used non-abelian finite p-groups in the literature. One is the group
1Research supported by a NBHM research grant.
724 Ayan Mahalanobis
of unitriangular matrices over a field and the other is the extra-special pgroups.
Weir [4] and Lev˘chuk [2](http://dx.doi.org/10.12988/ija.2013.3991) worked on the automorphism group of the
unitriangular matrices.


PS
I remember I have seen some papers with quite explicit description of automorphism, and I thought I collected such links in comments to my [old MO-question: Representation theory of p-groups in particular upper tringular matrices over F\_p](https://mathoverflow.net/q/106521/10446), but it cannot find it now again, hm-m... I'll try to find more...


