The Target Picker will attempt to filter out Action Target Groups that don't apply to the selected Character Set (either picked via the * button next to the drop-down box, or by being the only Character Set). The current version filters based on the main Character Set only -- this is getting fixed as of now to include the sets linked via Also Launch. You can de-select the Character Set by selecting the empty option in the list (via the * button)``, and then no filtering will be done. Every Character and Action Target Group will be listed.

But that should be irrelevant, since you've only got one Character on the other PC. You should see "Character:<Name>" for every Character you have, including the one on the other PC. Unless you're wanting to send something to a sub-set of all the characters that includes the one running on the other PC, and one or more but not all of the characters on the main PC (as in, you want to send something to 1 on the main PC plus the 1 on the second PC) then an Action Target Group is doing nothing special for you -- you can already pick Character:MyCharacterOnSecondPC as the target.