So we tend to talk about "the" orbit category of a group, but the orbit category construction is actually a functor.

If $G$ is a discrete group, then $\mathrm{Orb}_G$ is equivalent to a certain full subcategory of $\mathcal{S}_{/BG}$.

Any $f\colon BG\to BG$ induces a functor $f^*\colon \mathcal{S}_{/BG}\to \mathcal{S}_{/BG}$ by base change.

Which carries that full subcategory into itself.

So we actually have a functor $\mathrm{Orb}\colon B\mathrm{Aut}(BG) \to \mathrm{Cat}_\infty$.

Have people thought about the implicactions of this for equivariant homotopy theory?

For instance, does this idea extend to a functor $\mathrm{Mackey}\colon B\mathrm{Aut}(BG) \to \mathrm{StCat}_\infty$ (i.e., does the stable Mackey category of $G$ admit an action by $\mathrm{Aut}(BG)$?

A concrete example: $\mathrm{Aut}(B\Sigma_6) \approx \mathbb{Z}/2$, so $\mathrm{Orb}_{\Sigma_6}$ and hence $\mathrm{Top}_{\Sigma_6}$ admits a $\mathbb{Z}/2$-action.

What is $\mathrm{Top}_{\Sigma_6}^{h\mathbb{Z}/2}$?

What if $G$ is a Lie group? What is the natural thing that acts in that case?

Saturday morning thoughts ...

@CharlesRezk If $G$ is a Lie group, does one need conditions on the map $f: BG \to BG$ to ensure that pullback carries orbits to orbits?

It does seem very striking to me that the orbit category is really just a full subcategory of Borel $G$-spaces -- why that subcategory?

One can talk about incomplete families

what about "supercomplete families" which include some non-orbits as well?

But I guess that wasn't quite where you were going with this

One thing I would like to know is some characterization of the orbit category as a subcategory of $Top/BG$ which works for compact Lie groups and uses only homotopy-invariant / $\infty$-categorical concepts

Because the only way I know -- by saying the action is transitive -- seems inherently point-set based

Heck, for that matter I don't know an $\infty$-categorical characterization which I really like of the orbits for a discrete group.

I guess you can say they're the indecomposable 0-truncated objects

But I don't think that sheds much light on why they're important

Regarding the specific example -- $Out(\Sigma_6)$ lives on an island in my head. I don't know of any connections to other "exceptional phenomena" like octonions or monster groups or whatever. It seems like exceptional things are usually connected, so it's kind of strange that this one isn't. But maybe there's something to say homotopy-theoretically here...

@CharlesRezk Equivariant homotopy theory (like representation theory) takes groupoids as input, not groups. I'm not aware of any deep implications of this observation, but on a basic level it really is quite useful to understand various functors (eg, induction, fixed points, geometric fixed points, norms, ...) and "base change" properties between them.

@TimCampion presumably, since many maps $BG\to BG$ don't come from Lie group homomorphisms. I suppose that observation suggests the answer: for a Lie group $G$, you have a topologically enriched category whose objects are automorphisms $G\to G$, while morphisms between such are conjugation by appropriate elements of $G$. This is a Lie-group version of "$\mathrm{Aut}(BG)$"

@TimCampion There are indecomposable 0-truncated objects which are not equivalent to orbits.

@MarcHoyois Thanks.