Inner, outer, and total automorphism of $Spin$ groups

Question

The center Z$(G)$, the automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$) are related. They form short exact sequences: $$ 1 \to \text{Z}(G) \to G \to \text{Inn}(G) \to 1, $$ $$ 1 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 1, $$ and a combined exact sequence $$ 1 \to \text{Z}(G) \to G \to \text{Aut}(G) \to \text{Out}(G) \to 1. $$ If $G$ is a simply-connected compact Lie group and ${\bf{g}}$ is its Lie algebra (necessarily semi-simple), then $$\text{Inn}(G)=\text{Inn}({\bf g})=\mathrm{P}G \text{ (the projective group) },$$ $$\text{Aut}(G)=\text{Aut}({\bf g}),$$ 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}} $.

My question

What are the inner, outer, and total automorphism groups of $Spin$ groups? Say $Spin(n)$ for any positive integer $n$?

1. For $G=Spin(1) = \mathbb{Z}/2$, the $\text{Z}(G)= \mathbb{Z}/2$, $\text{Inn}(G)=0$, $\text{Out}(G)=0$, and $\text{Aut}(G)=0$, correct?

2. For $G=Spin(2)=U(1)=S^1$, the $\text{Z}(G)= G=U(1)$, $\text{Inn}(G)=0$, $\text{Out}(G)=\mathbb{Z}/2$, and $\text{Aut}(G)=U(1)$, correct?

3. For $G=Spin(3)=SU(2)=S^3$, the $\text{Z}(G)= G=\mathbb{Z}/2$, $\text{Inn}(G)=PSU(2)=SO(3)$, $\text{Out}(G)=0$, and $\text{Aut}(G)=PSU(2)=SO(3)$, correct?

4. For $G=Spin(n)$ with $n \neq 8$ and $n =0 \mod 4$ but $n>3$, the $\text{Z}(G)= \mathbb{Z}/2 \oplus \mathbb{Z}/2=(\mathbb{Z}/2)^2 $, $\text{Inn}(G)=PSpin(n) \equiv\frac{Spin(n)}{(\mathbb{Z}/2)^2}=\frac{SO(n)}{\mathbb{Z}/2}$, $\text{Out}(G)=\mathbb{Z}/2$, and $\text{Aut}(G)=PSpin(n) \rtimes \mathbb{Z}/2=\frac{SO(n)}{\mathbb{Z}/2} \rtimes \mathbb{Z}/2$, correct? How exactly is this semidirect product $ \rtimes$ defined?

5. For $G=Spin(n)$ with $n \neq 8$ and $n =2 \mod 4$ but $n>3$, the $\text{Z}(G)= \mathbb{Z}/4$, $\text{Inn}(G)=PSpin(n)\equiv\frac{Spin(n)}{\mathbb{Z}/4}$, $\text{Out}(G)=\mathbb{Z}/2$, and $\text{Aut}(G)=PSpin(n) \rtimes \mathbb{Z}/2=\frac{Spin(n)}{\mathbb{Z}/4} \rtimes \mathbb{Z}/2$, correct? How exactly is this semidirect product $ \rtimes$ defined?

6. For $G=Spin(n)$ with $n =1,3 \mod 4$ but $n>3$, the $\text{Z}(G)= \mathbb{Z}/2$, $\text{Inn}(G)=PSpin(n)\equiv\frac{Spin(n)}{\mathbb{Z}/2}=SO(n)$, $\text{Out}(G)=\mathbb{Z}/2$, and $\text{Aut}(G)=PSpin(n) \rtimes \mathbb{Z}/2=SO(n) \rtimes \mathbb{Z}/2$, correct? How exactly is this semidirect product $ \rtimes$ defined?

7. For $G=Spin(8)$, the $\text{Z}(G)= \mathbb{Z}/2 \oplus \mathbb{Z}/2$, $\text{Inn}(G)=PSpin(8)=SO(8)$, $\text{Out}(G)=S_3$, and $\text{Aut}(G)=SO(8) \rtimes S_3$, correct? How exactly is this semidirect product in $SO(8) \rtimes S_3$ defined?

(Note that $\pi_1(Spin(n=1))=0$, $\pi_1(Spin(n=2))=\mathbb{Z}$, $\pi_1(Spin(n \geq 3))=0$.)