Hvac Wood stove-Running ductwork within an air return?

[Daniel B.]

Hi,

  I have a wood burning stove that I am trying to circulate heat.  In the room is an air return that is basically just a double wall cavity from close to the ceiling to the furnace which is in the basement.
I would like to get some of the hot air from the stove  into the basement.  I would like to add a vent with a fan to suck air from the wall by the stove (on main floor) to the basement.  This is not easily done without tearing into walls.  Can I mount this 4” pipe basically at the grate of the air return and run it through this air return (just drywall) until it hits the basement ductwork?  I would then cut a hole in the ductwork the same size of the new vent and seal so this vent would not leak into air return. 
   is it against code to run this within air return?  Home is 5 years old and I have electric in an outlet next to the air return cavity so it seems like the easiest way to do it?  I would use electric to add some kind of a fan to force hot air from main floor into basement…