First, for generalities related to the question, I'd recommend the article of Bushnell and Henniart, " Generalized Whittaker Models and the Bernstein Center," in Amer. J. of Math., Vol. 125, No. 3, Jun. 2003, pp.513-547.  Starting in Section 8, you can find a nice and explicit survey and expansion on the results of Bernstein.  Some of the "what do I know" facts you mention can be found in this paper of Bushnell-Henniart (see Example 8.6).

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I started to write a complete answer here and accidentally hit the "Post" button.  It seems more difficult than I thought to cook up an example.  I'll try to think more about it, and I'll post at a later time/date.