Describe the characteristics that a pyramid must have for it to have an axis of rotational symmetry.

Answer 1

It must be a right pyramid (i.e. not skewed). A vertical dropped from the vertex must fall into the middle of the base.
The base must possess rotational symmetry.
The order (n) of rotational symmetry of the base decides that of the pyramid.
A rectangle or a parallelogram will give a rotational symmetry of order 2. An equilateral triangular base gives n=3, etc.
Read the following article which gives a lot more insight on the subject:
http://en.wikipedia.org/wiki/Rotational_symmetry

Answer 2

Thank you MathMate for your help and the link.