Let $G$ and $H$ be groups, both acting on a set $X$ on the left, in such a way that the two actions commute. (Equivalently, let $G \times H$ act on $X$.)


The set $\text{Fix}\_H(X)$ of $H$-fixed points carries a $G$-action, and we can then take the set $\text{Fix}\_H(X)/G$ of orbits. But also, the set $X/G$ of $G$-orbits carries an $H$-action, and we can take the set $\text{Fix}\_H(X/G)$ of fixed points. There is a canonical map
$$
\lambda\colon \text{Fix}\_H(X)/G \to \text{Fix}\_H(X/G),
$$
which is easily seen to be injective.


It's a fact that if $G$ and $H$ are finite groups with coprime orders, this map $\lambda$ is bijective. So then,
$$
\text{Fix}\_H(X)/G \cong \text{Fix}\_H(X/G).
$$
(A bit more generally, this is true whenever $G$ and $H$ are possibly-infinite groups with finite coprime exponents.) The proof isn't hard. 


I only discovered this yesterday, but I guess it's well-known — maybe as a special case of something more general. Can anyone give me a reference?


**Update** Perhaps I should have explained the context. In category theory, limits commute with limits and colimits commute with colimits, but limits do not usually commute with colimits. There are various theorems giving restricted conditions under which limits *do* commute with colimits. This result about group actions, which [Peter Johnstone](https://www.dpmms.cam.ac.uk/people/p.t.johnstone/) and I came upon, gives a new (?) such theorem. That's why I'm after a reference. For more explanation, see [this $n$-Category Café post](http://golem.ph.utexas.edu/category/2013/12/commuting_limits_and_colimits.html).



When taking the fixed points commutes with taking the orbits
To see when this is possible, we can consider each orbit of $G \times H$ separately. An orbit of $G \times H$ corresponds to a subgroup of $G \times H$.


A subgroup of $G \times H$ corresponds to a triple of $A,B,C,D$ where $A$ is a normal subgroup of $B$ a subgroup of $G$, $C$ is a normal subgroup of $D$ a subgroup of $H$, plus an isomorphism $B/A \cong D/C$. The subgroup corresponding to $A, B, C,D,\cong$ consists of pairs $(g,h) \in G \times H$ where $g \in B$, $h \in D$, and the class of $g$ in $B/A$ is sent by the isomorphism to the class of $h$ in $D/C$. One can also reverse this process.


$X/G$ is a single orbit of $H$. It is the orbit corresponding to the subgroup $D$, that is, the action of $H$ on $H/D$. So $X/G$ has an $H$-fixed point if $D = H$. 


On the other hand $Fix\_H(X)$ is nonempty if and only if $C= G$. In this case, the map is surjective. So my comment was off slightly - $H$ and $G$ have this property if and only if no nontrivial quotient of $H$ is a subquotient of $G$. This condition is quite messy in general, and there are lots of examples, like $G$ finite and $H$ not residually finite.


But this is certainly a general result that your statements follow from.


