Sorry - I'll clarify.

I'm a US citizen, but the person I am inquiring for is a Jamaican citizen who lives in Jamaica.

The invite from an Italian citizen is already taken care of, but I'm trying to understand the visa requirements other than that, and if the flight that gets booked from Jamaica to Italy happens to pass through the US, is there any involvement in US customs? I wouldn't think so, but I am not sure - especially after hearing there is something called a transit visa. And I don't understand what country issues the transit visa...