Recently, I stumbled upon a theorem by Wolfgang Gaschütz (see below) that every non-trivial $p$-group which is non-trivial has an outer automorphism. However, the proof uses cohomology theory and as far as I can see, the proof is non-constructive.
I have also found a refinement of this theorem by Schmid, another elementary proof by Webb and the proof in Huppert's book (all non-constructive to my eye).
Does anybody know whether there exists a sort of algorithm or at least a constructive proof of an outer automorphism?
Schmid, Peter, Normal (p)-subgroups in the group of outer automorphisms of a finite (p)-group., Math. Z. 147, 271-277 (1976). ZBL0307.20016.
Gaschütz, W., Kohomologische Trivialitäten und äußere Automorphismen von p- Gruppen, Math. Z. 88, 432-433 (1965). ZBL0199.06302.`