Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first.
Let $B$ be a topological space and $G$ a topological group acting on a space $F$. Then the space $G^B$ of continuous maps from $B$ to $G$ has a group structure given by pointwise multiplication. Moreover, for nice enough spaces the compact-open topology gives $G^B$ the structure of a topological group. This group acts on the space $F^B$ (again endowed with the compact open-topology) in the obvious pointwise way and the action is continuous.
My question is the following: assume $G$ acts transitively; can we infer the same holds for $G^B$? If not, under what assumptions is transitivity preserved?
Notice that freeness and faithfulness are (separately) preserved (it is enough to think pointwise).