Maybe I will just answer the specific question the OP is interested in by giving a reference of sorts (in case it is helpful to the OP).

In Bourbaki *Éléments de mathématique: groupes et algèbres de Lie.* Chapitre 9 [Exercise 3][1] one is asked to prove that $\mathrm{Out}(PSU(n))\cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$.

It seems to me that in this case one can show this explicitly.   The non-trivial non-inner automorphism is represented by the Cartan involution $A\mapsto (A^{-1})^T$ on $PSL(n,\mathbb{C})$ which simplifies to $A\mapsto \overline{A}$ on $PSU(n)$.


  [1]: https://www.google.com/books/edition/Groupes_et_alg%C3%A8bres_de_Lie/m_bKwNLBZk4C?hl=en&gbpv=1&bsq=%22cyclique%20d%27ordre%202%22