If I purchase a game on my accounts can my son play it on his accounts (same console)

Yes, you'll need to set the console as your Home console in order to share with him https://support.xbox.com/en-US/help/hardware-network/console/my-home-xbox

Does it matter who installs the game? Me, my oldest son, and my youngest son all have different accounts but we belong to the same family account.

My oldest son uses the Home console and signs into that console with his own account. My youngest son has his own console and signs into that console with a different account. If my oldest son installs the game on the Home console will my youngest son be
able to see the game on his own console while signed in with his account? Or does my oldest son have to sign on to both consoles?

SOLVED: I had the same problem and this answer solved it, for me. I wanted to install a Play Anywhere game (Hello Neighbor) on my sons Windows 10 account and could not, after reading this I opened the Windows Store App, signed out my son and did a temporary
login with my account (the same email account that purchased the game), now the install button was there. After having done the install I did a login with his account again. My son can now play the game using his account on our Windows 10 PC. Thank you
very much for providing a solution ArminatorX!

Since you are sharing Home with gold in your xbox one your son can still play the game purchased under your account if his account is present in the same console.

Hey AZ,

Thank you for posting on the forums today! I'd be happy to help.

You can share your digital games to only ONE console, and not two unfortunately. To set up the game sharing, you need to set up the console you want to share with as the

Home Console for the account that purchased the game.

Once that has been set up, other accounts on that same Xbox have the benefit of playing your games and using your gold subscription to play online, without having to be signed in as you. But as mentioned previously this can only be done on one console.

They only way I believe to make it work on the second console would be that you need to have your account always signed in, while your second son signs in to his account. Both accounts are signed in, but then you just switch the account that's playing and
he should be able to play the game on his console as long as your account is still signed in at the same time.

Let us know if you have any further questions.

Is this the same answer if I had a disc instead of a digital download? My oldest son has his own account and he uses the Home console, and my youngest son has his own account and he uses his own console. I have my own account and in the head of the family,
but I hardly ever play.