Let $X$ be a separated integral normal scheme of finite type over $\mathbb{C}$. It is my understanding that $\mathbb{Z}/2$ acts on the homotopy type of $X(\mathbb{C})$ and its profinite completion, $\hat{X(\mathbb{C})}$. Is it true that taking homotopy fixed points commutes with profinite completion (i.e. $\hat{X(\mathbb{C})^{h\mathbb{Z}/2}}\approx \hat{X(\mathbb{C})}^{h\mathbb{Z}/2}$)?

Context: I thought of this question after reading some discussion about Sullivan's conjecture and etale homotopy types and I am very bad in algebraic topology, so the question might be trivial.

Let $X$ be a separated integral normal scheme of finite type over $\mathbb{C}$. It is my understanding that $\mathbb{Z}/2$ acts on the homotopy type of $X(\mathbb{C})$ and its Sullivan 2-profinite completion, $\hat{X(\mathbb{C})}$. Is it true that taking homotopy fixed points commutes with profinite completion (i.e. $\hat{X(\mathbb{C})^{h\mathbb{Z}/2}}\approx \hat{X(\mathbb{C})}^{h\mathbb{Z}/2}$)?

Context: I thought of this question after reading some discussion about Sullivan's conjecture and etale homotopy types and I am very bad in algebraic topology, so the question might be trivial.

EDIT: an answer to this question suggests that for a trivial $\mathbb{Z}/2$-action, the answer to our question is positive. However, the action on the underlying topological space of the schemes we are considering is not trivial (in positive dimensions) so this is not directly relevant.