Raise the sash window up high enough to allow the pet door to sit comfortably in the gap. Measure the height of the gap and add 4 inches. This measurement will be the height of your plywood.
Measure the width of the window gap. This measurement will be the width of your plywood.
Cut the plywood to size using a square to get 90 degree angles.
Insert the pet door into the piece of plywood, following the instructions in your pet door kit; you should treat the piece of plywood as the bottom of a door for instruction purposes. It will involve cutting an opening for the pet door.
Place the mounted pet door in the window gap. Close the window onto the top of the plywood.
Measure the distance from the top of the opened window sash to the top of the window frame track. Cut two wooden dowels to size and and place the dowels in the window tracks to prevent the window from being opened from the outside. For a more permanent solution, screw two wood screws above the window (one on each side) into the tracks.