  In this paper we consider finite 2-groups with odd number of real conjugacy
classes. On one hand we show that if $k$ is an odd natural number less than 24,
then there are only finitely many finite 2-groups with exactly $k$ real
conjugacy classes. On the other hand we construct infinitely many finite
2-groups with exactly 25 real conjugacy classes. Both resuls are proven using
pro-$p$ techniques and, in particular, we use the Kneser classification of
semi-simple $p$-adic algebraic groups.
