Does the fixed point functor preserve colimit and limit?

Question

Let $G$ be a compact Lie group and $g\in G$. Let $C_G(g)$ denote the centralizer of $g$ in $G$. Consider the functor $X\mapsto X^g$ from the category of $G-$spaces to the category of $C_G(g)-$spaces.

Does this functor preserve colimits? Does it preserve limits?

Thanks.