There is a difference between "asked after them" (though that's rather colloquial) and "asked for them" in English, but Duolingo accepts either translation. I suggested "asked about them," which I'd consider synonymous with "asked after them" but got the big red X.
Which best conveys the sense of the German sentence?
"Ask after" and "ask about" would be something like "How is your father?" Or "How does your sister like her new job?" Of the two, "ask after" is a bit colloquial; some might consider it unacceptable.
"Ask for" would be more like "May I have Charles for my waiter?" It needn't even take the form of a question: "Please ask Charles to come out here.
Which of these--if any of these--would be the best translation here?
Here are the three phrases translated by Google -
"about them" - "... hat danach gefragt"
"for them" - "... hat darum gebeten"
"after them - "... hat nach ihnen gefragt"
After a little more thought I think that "after" implies both "for" and "about" simultaneously, as it implies a desire to see them which might not be fulfilled.
Apparently, in German it can be either. My German source says that he would understand this sentence either way, depending on context.
It would work equally well if the previous sentence were:
"Please ask his sisters to visit." [he was asking for them] or:
"Are his children well?" [he was asking about them] or:
"Please bring his shoes." [he was asking for them]
He does say that, in the last case, when what he has been asking for is a thing and not a person, you might also say, "Er hat danach gefragt".