And folding fins made out of...?
The classic materials would be galvanized metal; cartridge brass; and copper.
Rotary fins, which wrap around the projo, have am issue in that you need a way to get their hinges flush to the lands diameter, or they might get sheared off. They are relatively simple once the geometry is worked out. Bend them out of something "springy," set them in the projo, slide in the pins, then bind with a cord or band in a grove. They "stack" around the projo based on number; three would encompass a 120º arc (less a bit for the hinge); four, an arc of 90º, and so on.
Axial fins are a bit different. They fall into two rough categories, front hinged and back hinged. Axial fins sit in a matching longitudinal groove in the projo, or along a reduced diameter portion of the projo (proportionate to the width of the fins). The front hinge versions usually have a spring portion cut into or as part of the fin. The front edge of the fin is usually profiled so that it provides a stop when the fin hinges open. As with rotary fins, a band or cord holds them closed until launching.
The rear hinged fins are similar, with the back edge of the fin typically being the mechanical stop to hold them in the open state. US FFAR (folding fin aircraft rockets) work in this way, the fins pop open and index against the flare around the base of the rocket nozzle.
Now, a neat thing about rear-hinged fins. you can actually link the fins to the base of the projo and allow that base to slide a distance up the length of the end of the projo. Once free of the muzzle, the fin pop open, and air drag will keep them there. This is slick, as you don't need springs,
per se.