Yes, it is possible to have a square pyramid be made up of 4 isosceles triangles and one square as its base.
The only condition is that the base angles of the triangular faces can be less or more than 45 degrees, but never 45 degrees.
If the base angles are less than 45 degrees the triangular faces will be obtuse isosceles triangles, and if the base angles are more than 45 degrees the triangular faces will be acute isosceles triangles.
If the base angles of the triangular faces are 45 degrees each, there is no pyramid.
