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In The Punisher season 2, we see Billy Russo wearing this mask:

Who provided Billy Russo with this mask which totally signifies his character in this series (Jigsaw)?

If the hospital provided the mask why would they provide this kind of scary mask to wear?

It would be quite understandable if he himself purchased or created a mask to wear but as he is in a hospital under supervision I don't think he could get the mask by himself.

Billy decorates it himself, after Krista suggests it.

I don't think we're shown the actual event, but in episode four, Scar Tissue, there's a flashback scene to early in Russo's rehabilitation where he's wearing a plain white mask.

Whether it's to protect his face as it heals or to hide it to prevent his scars upsetting himself or others, we don't really know.

During the session, Krista is asking him about his dreams, seeing the skull [which he hasn't at this point figured out how/why he sees it].

This is the ensuing conversation...

Krista: There's this thing I do with my TBI patients. They make masks
that help them represent what they want to portray to the outside
world. You already have the mask, why don't you try it?

Russo: You want me to decorate this thing?

Krista: That's exactly what I mean.

Russo: Well, I'm not in kindergarten, Doc. And I ain't gonna finger
paint.

Krista: Just do me a favour and think on this. What face do you want
the world to see?

The overall scene is quite long & in two distinct parts. This part starts at around 14:30, the entire scene begins around 11:20.

The transformation is really quite clever. The plain mask has a distinctly sad, sympathetic face. Painted, it turns that into a Heath Ledger Joker-type smile & hides the eye shape in all black.