Pick one that goes well with the Groups and Plots you were dealt. Shuffle the cards after the last player has chosen. Each player puts his chosen Illuminati card on the table, all at once. See p. 13 of the INWO rulebook. Each player chooses one Group card from his hand, as the first puppet of his Illuminati.
All players show their choices at the same time.
Any puppets which are chosen by more than one player go back into the owners' hands, and they must make alternate choices. This continues until there are no duplicate puppets. Each player rolls two dice. The one with the highest roll goes first. You may not use Plot cards or special abilities on them, either.