Suppose $M$ and $N$ are manifolds, each acted on freely by a group $G$. If it helps, I'm happy to assume $M$ and $N$ are compact and $G$ is a finite group (or a compact Lie group). Consider the "diagonal" action of $G$ on the product manifold $M \times N$, defined by $g \cdot (m, n) := (g \cdot m, g \cdot n)$.
What can be said about the topology of the quotient?
In particular, when is $(M \times N)/G$ homeomorphic to a product of manifolds, and when is it homeomorphic to either of the products $M/G \times N$ and/or $M \times N/G$?
I suppose the Künneth theorem gives necessary conditions for the quotient to be a product. Are there other necessary and/or sufficient conditions?
Can more be said if $M = N$ and the action of $G$ is the same on both?
If not much can be said in general, I am especially interested in spheres and lens spaces. For simplicity, let's say $M$ and $N$ are spheres and $G = \mathbb{Z}_2$ consists of the antipodal map and the identity (so then $M/G$ and $N/G$ are each a real projective space).
For example, suppose $M = S^3$ and $N = S^5$. I think the Künneth theorem shows that $\mathbb{RP}^3 \times S^5$ and $S^3 \times \mathbb{RP}^5$ have different integral homology groups, so they are not homeomorphic. So in this case, $(M \times N)/G$ cannot be homeomorphic both to $M/G \times N$ and to $M \times N/G$.
We could also consider spheres of the same dimension. For example, if $M = N = S^1$, then I think (from drawing a picture) that the quotient is homeomorphic to the torus $S^1 \times S^1$. (But maybe this is just luck due to the low dimension. Note $\mathbb{RP}^1 = S^1/\mathbb{Z}_2$ is just $S^1$ again.) If $M = N = S^3$, is the quotient homeomorphic to $\mathbb{RP}^3 \times S^3$? What if only one of the factors is $S^1$?