TITLE: Decomposition of induced representations / Refinement of Mackey's criterion
QUESTION [14 upvotes]: There are already some questions with almost the same title, but they are more restrictive.
Let $G$ be a finite group, $H$ a subgroup, $V$ an irreducible representation of $H$,
and $W=Ind_H^G V$ the induced representation. 

Is there a simple way to compute the number of irreducible components of $W = Ind_H^G V$?

Let me precise my question by saying what kind of answers I would hope for. 
There is a well-known description of $Res_G^H Ind_H^G V = Res_G^H W$ as follows. Let $S$ be a set of representatives of the double classes of $H \backslash G / H$. For $s \in S$, let $H^s$
be the group $sHs^{-1} \cap H$ and $V^s$ the representation of $H^s$ on the space $V$ where $x \in H^s$ acts by $s^{-1}xs$ on $V$. Then
$$Res_G^H W = \oplus_{s \in S} Ind_{H_s}^H V^s.$$
Using this description, one gets using Frobenius reciprocity twice that $$Hom_G (W,W) = Hom_H(V,Res_G^H W) = \oplus_{s \in S} Hom_H(V,  Ind_{H^s}^H V^s) = \oplus_{s \in S} Hom_{H^s} (V,V^s)$$
which implies the well-known criterion of Mackey: $W$ is irreducible if and only if for all $s \in S$, $V$ and $V^s$ have no $H^s$-irreducible subrepresentations in common.
So assuming we can compute $S$, the groups $H^s$, the representations $V^s$, and how $V$ and $V^s$ decomposes as representations of $H^s$ (this may be hard in practice, but let's assume we can do that), then we know when $W$ is irreducible, and in general we know how
to determine the dimension of $Hom_G(W,W)$. (Of course all of this is standard cf. Serre Linear Representations for instance). 
Yet this falls short of answering the question,
because if for example you know that the dimension of $Hom_G(W,W)$ is $4$, that doesn't tell you if $W$ is the sum of 4 non-isomorphic irreducible representations, or of 2 copies of the same irreducible
representation. What I'd like would be a way to tell us, by looking at the $H^s$ and the $V^s$ etc, of determine which one it is. Or more generally, how to "read" the decomposition of $W$ into irreducible reps. in terms of the $V^s$, $H^s$ etc. If it is not possible, I would also 
like an explanation why.

REPLY [2 votes]: The proof of Mackey's theorem on intertwiners actually tells you how to construct the endomorphism algebra of an induced representation, not just its dimension. So, if you work a little harder, you may be able to get the additional information that you want.
To see how this algebra can be found, you may refer to my notes
http://www.imsc.res.in/~amri/html_notes/notesch1.html#x4-70001.4
A convolution product can be defined on $\Delta$'s in the notes, which will correspond to multiplication in the endomorphism algebra.
Added:
An interesting special case is decomposing parabolically induced representations of general linear groups over finite fields, which is beautifully explained in the notes of Howe and Moy Harish-Chandra homomorphisms for $p$-adic Groups.
Such algebras are often called Hecke algebras, and there is a vast literature on them.