  For an action of a finite group on a C*-algebra, we present some conditions
under which properties of the C*-algebra pass to the crossed product or the
fixed point algebra. We mostly consider the ideal property, the projection
property, topological dimension zero, and pure infiniteness. In many of our
results, additional conditions are necessary on the group, the algebra, or the
action. Sometimes the action must be strongly pointwise outer, and in a few
results it must have the Rokhlin property. When the group is finite abelian, we
prove that crossed products and fixed point algebras preserve topological
dimension zero with no condition on the action.
  We give an example to show that the ideal property and the projection
property do not pass to fixed point algebras (even for the two element group).
The construction also gives an example of a C*-algebra which does not have the
ideal property but such that the algebra of 2 by 2 matrices over it does have
the ideal property; in fact, this matrix algebra has the projection property.
