I asked this question on MSE https://math.stackexchange.com/questions/4475382/extension-of-an-involution-on-g-to-an-involution-on-g-mathbbc but didn't receive any answer so far. My question is the following:


Let $G$ be a compact connected Lie group and  $ \sigma :G \rightarrow G $ be an involution on $G$. Let $G^\sigma :=\lbrace g \in G, \sigma(g)=g \rbrace$.
 
Denote by $G_\mathbb{C}$ the complexification of $G$. Does there exist an involution $\tilde{\sigma}$ on $G_\mathbb{C}$ which coïncides with $\sigma$ on $G$ and such that ${(G^{\sigma})}_\mathbb{C} = {(G_\mathbb{C})}^\tilde{\sigma}$ ?

My though was to define $\tilde{\sigma}$ to be $\sigma$ on G and to be the identity on  the complement of $G$ on $G_\mathbb{C}$, but don't see whether this imply the ${(G^{\sigma})}_\mathbb{C} = {(G_\mathbb{C})}^\tilde{\sigma}$ or not, any help please!