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.

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.