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.