Can I work in a co-op this summer as an international student?

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Hi everyone,
I’m an international student with a bit of a tricky situation, and I could use some advice.

I’m graduating this May from my current institution, and I’ll be starting a new master program at a different institution in September. Of course, I’ll need to apply for a new study permit for my new program.

Here’s the thing: I’ve got a co-op offer for this summer (4 months), but I don’t want to use my Post-Graduation Work Permit (PGWP) for it since I want to save it for after my future studies.

Can I still work this summer? My questions are:

Can I use my current study permit and co-op work permit (if valid) for this co-op?

If I apply for a new study permit for September, would I still be allowed to work this summer?

Is there a way to do this without activating my PGWP?

Thanks in advance for your help!