After a few minutes, the family saw an ellipsis next to Faisal’s icon, indicating he was replying.

“I understand you may consider us being radical Muslims whatever that maybe? Well please don’t believe everything on fox news!!” he writes, his message riddled with typos. “We don’t agree with terrorism AT ALL … You have my word but also the word of her friends IN NO way will we ever try to make her harm others or do anything which is illegal.”

She typed: “Nothing you say explains the offer of a trip to Austria, the free ticket, the offer of a marriage deal with an old, bald man.”

He replied that the marriage offer was “a joke.”

Then he gave his word he would not contact Alex again.

Afterward, Alex agreed to hand over the passwords to her Twitter and email accounts. Her grandmother changed them to prevent her from using them. Federal Bureau of Investigation agents came to the house and downloaded her electronic communication history after Alex’s family contacted the agency. An F.B.I. official at state headquarters would not discuss details of the case.

In an email to the family, one of the agents later wrote that no one wanted “to see her get caught up in any of the dangers that she has been extremely close to in the past.” But he acknowledged that Alex was also under scrutiny, saying the agency’s goal was not just to keep her safe, “but also the rest of the country safe as well.”

After the online showdown with Faisal, Alex and her grandparents left for a much-needed vacation in their recreational vehicle, seeking to reconnect.

Alex found she could not stay away from her online friend for long, though. Even though she had come to feel she couldn’t trust him, she still missed his companionship.