You could ask President Kennedy some questions relating to his religion. For example, you might ask him why he thinks Americans were finally able to accept a Catholic President in 1960. You might also ask him if being the first Catholic American President hindered him in anyway during his presidency.
You could also ask him about the 1960 Presidential Election. The 1960 election was one of the closest presidential elections in American history, so you could ask him what he thinks finally swung the election in his favor. In addition, there is some doubt over the legitimacy over the 1960 election, particularly over the votes in the state of Illinois. You could ask him he could shed some light on what some have called voter fraud.
You might also want to ask him some questions pertaining to the civil rights movement. JFK took a cautious approach to the civil rights movement initially. You could ask him why he took this approach at the beginning and what the catalyst was that changed his mind....
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