2-Groups with a fixed number of real conjugacy classes

We study the finite 2-groups with a fixed number of real conjugacy classes. The order of such groups can be arbitrarily large but we show that it can be bounded if the orders of the elements in a generating set are also fixed. If the number k of real classes is odd we show that the group order can be bounded in terms of k and the nilpotency class although we conjecture that a bound in terms only of k   exists. We confirm this conjecture when k=7k=7.