Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$

Question

Given a group $G$, we denote the center Z$(G)$, we like to know the automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences: $$ 1 \to \text{Z}(G) \to G \to \text{Inn}(G) \to 1, \quad \text{and } \quad 1 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 1, $$ and a combined exact sequence $$ 1 \to \text{Z}(G) \to G \to \text{Aut}(G) \to \text{Out}(G) \to 1. $$

Question:

For $G=U(N)$ as a unitary group, a subgroup in GL$(N, \mathbb{C})$ represented by rank-$N$ unitary matrices of complex coefficients $\mathbb{C}$.

We know the center $Z(U(N))=U(1)$.

What are the follows:

• Aut($U(N)$) = ?

• Inn($U(N)$)= ?

• Out($U(N)$)= ?

For $G=U(N)/\mathbb{Z}_k$ where $\mathbb{Z}_k$ is a finite abelian group of order $k$, which requires $k \vert N$ to be factorized.

We know the center $Z(U(N)/\mathbb{Z}_k)=U(1)$.

• Aut($U(N)/\mathbb{Z}_k$) = ?

• Inn($U(N)/\mathbb{Z}_k$)= ?

• Out($U(N)/\mathbb{Z}_k$)= ?

Additional background info (warm-up, which I had obtained):

If $G$ is a simply-connected compact Lie group and ${\bf{g}}$ is its Lie algebra (which would necessarily be semi-simple), then $\text{Inn}(G)=\text{Inn}({\bf g})=\mathrm{P}G$, $\text{Aut}(G)=\text{Aut}({\bf g})$, and $ \text{Out}(G)=\text{Out}({\bf g})= \text{Aut}(D_{{\bf{g}} })$ is isomorphic to the automorphism group of the Dynkin diagram $D_{{\bf{g}} }$ of the Lie algebra ${\bf{g}} $.

• For $G=U(1)$, we have $\text{Z}(G) =U(1)$, $\text{Inn}(G) = 1$, $\text{Aut}(G)=\text{Out}(G) = \mathbb{Z}_2$.

• For $G=SU(2)$, we have $\text{Z}(G) =\mathbb{Z}_2$, $\text{Inn}(G) = SO(3)$, $\text{Aut}(G)=PSU(2)=SO(3)$, and $\text{Out}(G) = 1$.

• For $G=SU(N)$ with $N$ $\geq 3$, we have $\text{Z}(G) =\mathbb{Z}_{N}$, $\text{Inn}(G) = PSU(N)$, and $\text{Out}(G) = \mathbb{Z}_2$. We also have $\text{Aut}(G)=PSU(N) \rtimes \mathbb{Z}_2$.

p.s. An earlier version of a Math.SE question has not reached a conclusion.