Add 4 inches to the finished height and width of the cutout. Transfer these measurements to the wall with a measuring tape and mark the corners of the cutout on both sides of the wall. Snap a chalk line across the corners to create an outline of the cutout.
Check the outline with a carpenter's square to ensure that the corners form 90-degree angles. Cut through the drywall along the outline on both sides of the wall with a reciprocating saw. Use a sledgehammer to remove the drywall to form the opening.
Cut the wall studs at the top and bottom of the opening with a reciprocating saw. Remove the wall stud sections from the opening.
Cut two sections of 2-by-4 inch pressure-treated lumber to the width of the opening with a circular saw. These sections form the top and bottom plates of the opening. Attach the top and bottom plates to the cut ends of the wall studs with 4-inch wood screws and an electric screwdriver.
Cut two sections of 2-by-4 inch pressure-treated lumber to the interior distance between the top and bottom plates with a circular saw. These sections form the side studs of the opening. Stand one of these sections on each end of the bottom plate.
Insert 4-inch wood screws through the side studs into the top and bottom plates with an electric screwdriver. Drive the screws at 45-degree angles to secure the studs to the plates.
Cut sections of drywall with a utility knife to cover the plates and studs. Install the drywall sections with drywall screws at 4-inch intervals.
Tape over the drywall corners and seams with drywall tape. Spread drywall compound over the tape and screw heads with a putty knife. After the compound dries, sand the drywall surfaces with a palm sander.