$\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSO{PSO}$If I understand correctly, $\Aut\Spin(𝑛)$ automorphism of Spin groups splits as a semidirect product $$\Aut\Spin(𝑛) = \Out\Spin(𝑛) \ltimes \Inn\Spin(𝑛).$$ (This is not in general true for automorphism groups, although I am not sure about the situation for Lie groups specifically).

The inner automorphism group is given by $\Spin(𝑛)/𝑍(\Spin(𝑛))$, which equals the projective special orthogonal group $\PSO(𝑛):= \SO(𝑛)/Z(\SO(𝑛))$

The outer automorphism group is $\Out\Spin(𝑛)≅\mathbb{Z}_2$ when $n \geq 4$. When $n=3$, $\Out\Spin(𝑛)≅0$. When $n=8$, then the $\Out\Spin(𝑛)≅S_3$.

The definition of Pin group is given in https://en.wikipedia.org/wiki/Pin_group.

My question is that what is

the outer and inner automorphism group of $\Pin^{+}(n)$ and $\Pin^{-}(n)$ groups?

the outer and inner automorphism group of $\Pin^{+}(n,1)$ and $\Pin^{-}(n,1)$ groups?

pinningof $G$ gives a splitting. See, e.g., Brian Conrad, Proposition 1.5.5. $\endgroup$simply connected simplecompact $\Bbb R$-group, see Borovoi and Evenor, Lemma 4.1. $\endgroup$13more comments