TITLE: Yoga of six functors for group representations?
QUESTION [22 upvotes]: I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can assign a triangulated category $D(BG)=D(BG,k):= D(Rep(G))$ the (perhaps bounded) derived category of the abelian category of representations of $G$ (in vector spaces over $k$ algebraically closed of characteristic 0). As far as I understand the formalism of six functors should translate (if at all) as follows:
If $\pi : BG \to pt$ then: 

$\pi_{!}(-)$ is a derived version of $(-)^G$ (invariants) = group cohomology. 
$\pi_*(-)$ is a derived version of $(-)_G$   (coinvariants) = group homology .

Generally if $\pi: BH \to BG$ then:

$\pi_!(-) = \mathbb{L}Ind^G_H(-)$, 
$\pi_*(-) = \mathbb{R}Coind^G_H(-)$
$\pi^*(-) = \mathbb{L}Res^H_G(-)$.

I have several questions questions:


Are there any mistakes/wrong intuitions in the above?
I'm missing the upper shrieks $\pi^!$ and as such also the duality
  functor. How do these look in general? What is the "dualizing
  representation" of a group? Is it something familiar?
How much of this carries over to the category of algebraic stacks (on the big etale site of schemes) $BG$ for $G$ linear algebraic group? (suppose all this happens over some fixed field for simplicity).

REPLY [3 votes]: I'm more familiar with sheaf cohomology than group cohomology, but I think this is at least partly an issue of notation, as Peter McNamara suggests. I will also assume that all groups here are discrete.
To expand on his answer a bit, your notation for the adjoint triple induction $\dashv$ restriction $\dashv$ coindunction (or extension $\dashv$ restriction $\dashv$ coextension) appears to follow the common algebraic geometry convention for ring maps, but they do that because affine schemes are the opposites of commutative rings, and left adjoints are swapped for right adjoints when you take the opposite category. 
So to make group cohomology look more like sheaf cohomology, the notation should be $\pi_! \dashv \pi^* \dashv \pi_*$. Then group cohomology is also right derived pushforward $\mathbf{R}\pi_*$, just like sheaf cohomology, and group homology is derived lower shriek, just like compactly supported sheaf cohomology.
Hence, Verdier duality should be given by an adjunction $\pi_! \dashv \pi^!$, but for group reps $\pi_!$ aleady has a right adjoint $\pi^*$, so for discrete group representations we are always in the Wirthmüller context where $\pi^!=\pi^*$.