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.