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.