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I should note now that I've tinkered around a bit more, that everything works as intended if I use the "-i identity_file" switch when connecting from "Automation" and point that at the private key on the "Automation".
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There are three systems involved here.
Numbers 2 and 3 are on the same network, #1 is on a remote network. #3 does not have a monitor attached, it is managed via SSH from #2.
Now #2 has public/private keys available for, and is able to connect to #1 and #3 without a password just fine.
I need to connect to #1 from #3 in the same way. I first tried copying the keys #2 uses for #1 over to #3. I'm prompted for a password when I attempt to connect to #1 from #3. I've used ssh-keygen while logged into #3 and imported those keys to #1 just as I did when setting up #2 to connect to #1. I'm still prompted for a password.
Why am I being prompted for a password when connecting from #3 and not for #2 ?
The only difference I can think of is, when connecting from #3, I'm already remotely logged into #3 from #2.