Automorphisms of $G/Z(G)$ with $G$ simply connected

Question

Let $G$ be a simply connected (if necessary, compact Lie) group with finite center $Z$ and $p:G/\to G/Z$ be the canonical projection. Is there any way to know if every element in $\operatorname{Out}(G/Z)$ lifts to an element in $\operatorname{Out}(G)$?

I'm trying to understand who is $\operatorname{Out}(SU(3)/\mathbb{Z}_3)$.

Take $\varphi\in\operatorname{Out}(G/Z)$. From the fact that $G$ is simply connected we have that $p\circ \varphi$ lifts to a homomorphism $\tilde{\varphi}:G\to G$, but with kernel $Z$. Is there any way to turn this $\tilde{\varphi}$ into an automorphism?

Answer 1

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 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 $PSU(n)$.