Consider both safety and planning points before beginning this project. Most exterior walls have studs spaced every 16 inches on center and most electrical lines will be stapled to the sides of studs. If you have fiber glasss insulation it is wise to wear a dust mask to protect your lungs.
Find the location of the studs in the wall, using the stud finder. Hold the stud finder over the wall, while moving it over the wall. The stud finder will light up or sound an alarm when the machine is placed over a stud. Mark the location of the studs on your interior wall with a pencil where you plan to place your dog door.
Cut the drywall about 3 inches in from where you marked a stud using the hole saw. This cushion of space will help prevent you from hitting an electric line when cutting. Finish cutting out the rectangular area that is needed, using the measurements of your dog door as a guide. Reserve a large piece of the drywall for later.
Remove the insulation from inside of the wall and place it in a garbage bag, reserving some insulation for later. Mark the inside exterior wall of the house to the desired size of your dog door. From the inside of your house, drill through the wall at each corner of this marking. This helps mark and start the location for your reciprocal saw, which you will be doing from the outside of your home. Keep in mind that your inside hole will be larger than your outside hole.
Connect the drilled holes using a straight edge and a pencil. Cut out the pattern with a reciprocal saw carefully. You will now have a rectangular hole drilled completely through from the outside wall of your house to the inside wall of your house.
Create a frame for your new dog door. Cut the 2 by 4's or 2 by 6's to fit your wall, depending on what type of stud you currently have installed. One piece will fit horizontally between the two studs and the other piece of will fit from the bottom of the horizontal piece to the top of your footer forming an off-center "T" frame. Nail these two pieces together from the top of the horizontal piece down into the side of the vertical piece using two nails. Making an off-center "T" frame will ensure that the frame will not block the opening of the dog door.
Install the "T" frame in the wall as one unified piece. Once you place the frame in the wall, you will need to nail the horizontal piece to the studs to the left and right of it, at a diagonal angle. Then, nail the vertical piece at a diagonal angle into the footer. Check to make sure that the frame is solid and secured to the studs.
Measure and cut a piece of the drywall that you reserved to fit any leftover wall space. Before installing the drywall, stuff the empty space with some of the insulation that you reserved earlier. Tape and texture the new joint with drywall spackling using the putty knife and drywall joint tape. Allow to dry thoroughly.
Install the dog door according to the manufacturer's specifications. The dog door should have an exterior seal and an interior seal. Before installing the exterior seal, apply a bead of caulk to seal any drafts between the sides, top and bottom of the dog door and the wall. Your pet can begin enjoying the dog door as soon as the caulk is dry.