In this situation, you are referring to the state of the person when she arrives at her home, which requires an adjective, "safe." It's a subtle difference, but if you say "Get home safely," you are referring to the method by which she travels, or "gets."
Example: "Jane arrived home safe by taxi."
"Safe" refers to Jane's condition when she arrived. She was all in one piece, as good as when she left.
Example: "Jane arrived home safely by taxi."
"Safely" refers to how Jane arrived. There were no incidents or accidents. The taxi got her home safely.