Classification of automorphism groups of groups of order $p^4$

Here's a description of the groups of order $p^4$. It's kind of primitive and explicit, but it's not too long.

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!

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.