Experiencing a gray screen followed by a restart, as well as the appearance of a folder icon with a question mark, can indicate a potential issue with your MacBook's startup process. There are a few troubleshooting steps you can try to resolve this problem:
1. Reset the System Management Controller (SMC): The SMC controls various hardware functions on your Mac, and resetting it can sometimes resolve startup-related issues. To reset the SMC, follow the instructions specific to your MacBook model. You can find the steps on Apple's support website.
2. Check the startup disk: The appearance of a folder icon with a question mark suggests that your MacBook is unable to locate a valid startup disk. Verify that your MacBook's startup disk is set correctly. Go to "System Preferences" > "Startup Disk" and ensure the correct disk is selected.
3. Run Disk Utility: Use Disk Utility to check and repair any disk-related issues that might be causing the problem. Restart your MacBook, and immediately press and hold the Command + R keys to enter the macOS Recovery mode. From there, choose "Disk Utility" and run First Aid on your startup disk.
4. Reset NVRAM/PRAM: Resetting the NVRAM (non-volatile random-access memory) or PRAM (Parameter RAM) can help resolve various startup issues. Restart your MacBook and immediately press and hold the Command + Option + P + R keys until you hear the startup sound for the second time.
5. Safe Mode: Boot your MacBook into Safe Mode to check if any third-party software or extensions are causing the problem. Restart your MacBook and hold down the Shift key immediately after hearing the startup sound until the Apple logo appears.
6. Reinstall macOS: If the issue persists, you might consider reinstalling macOS. This process will install a fresh copy of the operating system without affecting your personal files. Make sure to back up your important data before attempting this step.
Further, you can also check the below link for solutions to fix flashing question mark with folder on Mac:
https://www.stellarinfo.com/blog/repair-mac-flashing-question-mark-folder-error/
If none of these steps resolve the issue, it's recommended to contact Apple Support or visit an Apple Authorized Service Provider for further assistance. They can provide more specific guidance based on the exact model and symptoms you're experiencing.