In addition to the hypocaust under the floor, the wall was lined with flue tiles.
If the flue is damaged (the most common problem is deteriorated mortar between the sections of flue tile), have a professional make repairs or install a replacement flue liner before the fireplace is used.
Ideally this should be done in the spring, so if you discover damage, like cracked flue tiles, there will be plenty of time for repair work.
It is best to simply remove the extension and replace it with lighter materials, with special metal flue replacing the flue tile and a wood structure replacing the masonry.
A cap, hood, or shroud serves to keep rainwater out of the exterior of the chimney; rain in the chimney is a much greater problem in chimneys lined with impervious flue tiles or metal liners than with the traditional masonry chimney, which soaks up all but the most violent rain.
The trouble with these improvised rigs is that they create bigger problems if they damage the flue tiles.
If the flue is damaged - deteriorated mortar joints between sections of flue tiles are the most common problem - repairs must be made or a replacement flue liner installed before using the fireplace.
If the top of the chimney terminates in a square or rectangular flue tile, the fireplace is probably a masonry one.