  We explore some of the special features with respect to Bredon cohomology of
groups having all its finite subgroups either nilpotent or p-groups or cyclic
p-groups. We get some results on dimensions and also a formula for the
equivariant Euler class for certain groups. We consider the generalization for
Bredon cohomology of the properties of being duality or Poincar\'e duality and
study their behavior under p-power index extensions with coefficients in a
field of characteristic p.
