Let $G$ be a group and $X \to Y, X \to Z$ morphisms of $G$-sets with pushout $P$. Is then $P^G$ the pushout of $X^G \to Y^G, X^G \to Z^G$? This is not clear from general category theory, because pushouts do not have to commute with equalizers. I think it's true when $X \to Y$ is injective; my proof is a fiddly computation, perhaps someone can provide a nice proof?
 
Now let's work with $G$-spaces instead of $G$-sets. Then it is not clear at all to me if the canonical map from the pushout to $P^G$ is open. Is it true when $G$ is finite? Or if $X,Y,Z$ are $G$-CW-complexes? By the way, this was used in a lecture about how to compute the homology of the classifying space $BG$ with the help of $\underline{E} G$.