I reserved an overnight permit for July starting on a Thursday and exiting on a Saturday with my four friends. I am still able to make it to pick up the permit, but they have had work hold them up and can't start hiking until Friday. If my permit allows me to hike Thursday to Saturday, are there any issues with entering on Friday with them instead of the Thursday date as long as I have my permit?

As per the Inyo National Forrest website, "Everyone in your group must start on entry date and trail listed on your permit".

https://www.fs.usda.gov/main/inyo/passes-permits/recreation

Unfortunately the two most important features of hiking permits are the entry date and trailhead where your trip starts.

If slots show up or are currently available for your Friday start date, you might be able to change the details of the permit and change the entry date. Not sure if you can do that online, but calling the wilderness permit office would certainly work.

If they could make it late Thurs night you could get up to Lone Pine Lake which would mean you'd entered the wilderness on the appropriate date.