I came across this side hat recently. What do you think of it?

It is of very poor/cheap manufacture with threads coming loose along all of the hems. There is no stamp or mark inside except the size - 6 7/8. Also, as you can see the badge is on the side. The bottom prong is broken so the badge has been sewn on as indicated by the arrows. The buttons are white metal and made in Paris. They have a makers name Perfect??????? (can’t make out the rest) in a belt and buckle design.

Any ideas as to the authenticity of this cap and units that wore it? All comments gratefully received and appreciated.

Many thanks for your help

James