First!

If one knows something, one believes it, has justification for believing it, and it's true.

For one to know something, one must "be aware of the truth or factuality of" this knowledge. By implying that one "knows" something, they are, by definition, implying that a person has just reasoning for believing so. Assuming that people are rational in their acceptance of knowledge, as they should be assumed to be, would confirm the final statement in this sentence, "it is true", because to truly "know" something, one must have proper justification. Proper justification should be derived from factual truth, as the definition of "to know" states that it must.

The argument that someone may know something false is a misuse of the English language, rather than the assertion that their knowledge is indeed knowledge. "Knowing" something false is merely a belief, rather than actual knowledge of something falsely. When a child "knows" Santa Claus is real, we as adults "know" that this knowledge is simply a belief. This is no different for adults than it is for children. Up until around the 17th century, the vast majority of the population of the world believed, falsely, that the Earth was flat. They treated this as knowledge, but we in the present, with a much better understanding of our world, know this was a false belief without any real justification and obviously without factuality.

Therefore, to be able to say that one truly knows something, one must have proper justification, including factuality and truth rather than internal justification which produces beliefs, rather than true knowledge.