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If I read the post correctly, it is a cat her MIL has adopted, not a true stray; killing an animal that one's close relative considers their pet is likely to lead to a lot of family conflict. Also, the evidence does not sound like it was likely that the cat was the one who killed the birds. There is no point in killing something that is not the cause of the destruction. It has no affect on the situation, and can make things worse, not better.