It's not a good sentence. We would say that she "took her mother to the airport." Not that she "went to the airport to take her mother."
The identity of the mother is ambiguous. If Mary (note the spelling) and Kate are sisters, then it would be "their" mother. If it is really supposed to be the mother of only one "her" then it is unclear from just this sentence whose mother they are talking about. It could be Mary, Kate, or some other person.
"Mary and Kate took Kate's mother to the airport" would be a clear way to write, to give one example.
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