Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?

I would start with Bob Oliver's Whitehead groups of finite groups. LMS Lecture Note Series, 132. Cambridge University Press, Cambridge, 1988. It's complicated and technical, but there are a bunch of examples computed in the last section that might answer your question.
– Danny RubermanJul 24 '13 at 1:52

Ok. Thanks for the reference.
– W. PolitarczykJul 24 '13 at 13:41