iTunes authorization for Home Sharing Recently got new iPhones.

Due to update issues, I'm having to use a different computer to sync 4 different devices.

I have Home Sharing turned on, so I have access to all the apps stored on iTunes on our original Home Computer.

I'm trying to transfer apps into my computer's iTunes from the Home Computer so I can place these apps on our new phones. However, almost every time I try to transfer an app that's available on the Home Computer, my computer says I'm not authorized.

I have 4 different accounts authorized on my iTunes.

I've read different responses and they say to deauthorize iTunes and then re-authorize it.

I think the problem may be that all of these apps were purchased by different family members. However, when I try to authorize the computer under the account I know purchased the app, I'm still told I'm not authorized.

The Home Sharing account is different than the accounts that purchased the apps, could that be the problem? 

I'm just at a loss on how to fix this because it's not letting me transfer the apps from the Home Computer. Any suggestions?
I'm not sure whether your approach is fundamentally wrong, or not, but I've never done it that way - this has always been my approach

To get another user's purchase to that iTunes account, you need to go to Apps > App Store > [scroll right to the bottom] Purchased > Apps > [User Name]