Classification of automorphism groups of groups of order $p^4$ For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of p-groups of order $p^4$.
I duly searched the web for a while and all the group theory manuals I could lay my hands on but I didn't find anything, not a hint.
Could someone provide a reference?
I already posted the same question on maths stack exchange and Professor Holt guessed that there is no publication on the subject. I repost here to verify his guess thoroughly.
A good reference for the classification of groups of order $p^4$ would be useful too, since I found the following thesis On p-groups of low power order but a different reference to help me compare the possible approaches to the problem would be of great help!
 A: Here's a description of the groups of order $p^4$.  It's kind of primitive and explicit, but it's not too long.
A: There is a description of the automorphism groups of groups of order $p^4$ in the Thesis by B. Girnat: "Klassifikation der Gruppen bis zur Ordnung $p^5$." Staatsexamensarbeit, TU Braunschweig, 2003, whose advisor was B. Eick.
The description is given in terms of the action on a set of generators of the respective group.
If I may add a personal note, I have been searching for this information for a long time, and I found it incredible that such a basic and old topic had no treatment whatsoever published in any form on the web. But I was assuming that anything interesting can be found in English (either written or translated into it). This assumption has been proven wrong, and I state this here as a warning for the newbies like me!  
A: Well, the simplest reference is groupprops wiki. This paper by Behravesh and Mousavi proves some results, but for the classification refers to the recent work by Burnside (1897), so you might want to look there.
A: This has the content of a comment, but is somewhat longer and with more formatting than a comment allows, so I post it as an answer.
Anyway:  I tried the GAP computation for 2^4, and got results.  For 3^4, it already runs out of memory with the default settings.  It seems likely that pumping up the amount of memory available (with the command line option -o) might allow the computation to finish for $p$=3 and perhaps 5, but it doesn't look like you'll be able to calculate everything for a large number of $p$'s.
The StructureDescription command I'm using below is also fairly inefficient, and there might be better ways to understand the automorphism groups.
For $p$=2, the command and output are:
List(AllSmallGroups(Size, 2^4), x->StructureDescription(AutomorphismGroup(x)));
[ "C4 x C2", "(C2 x C2 x A4) : C2", "(C2 x C2 x C2 x C2) : C2", 
  "(C2 x C2 x C2 x C2) : C2", "C2 x D8", "C2 x D8", "(C2 x D8) : C2", "C2 x D8", 
  "(C2 x D8) : C2", "(((C2 x D8) : C2) : C3) : C2", "(((C4 x C2) : C2) : C2) : C2", 
  "(((C2 x C2 x C2 x C2) : C3) : C2) : C2", "C2 x S4", "A8" ]

