Bill,
Ah, the two different time limits present a problem. I'm quite certain there is no way to set a separate time limit when 'ignore time limits' is enabled for a role.
Using Groups does not solve the problem either because both groups still see the same exam. The only thing Groups will do for a quiz activity is allow the teacher to view the group results separately.
The real (but complicated) solution to this is to use Groupings of Groups. Then you create two different quiz activities (each with a different time limit), but each grouping will see only one quiz - the quiz for their grouping. You still have the problem of the book link to the quiz. To make this really work, you would then have to create two books, one for each grouping, with a link to the appropriate quiz activity.
Groupings of Groups is very powerful, but can get a little messy. I've used it for a music course that had Men's Chorus, Women's Chorus and Mixed Choir. We were able to tailor what each student could see in the course by creating various Groupings of the three groups.
It will be interesting to see if anyone else has a suggestion.
Floyd