The 32-year-old was killed outside a semi-detached house in Crawley Down, a village in West Sussex, on Sunday.
She was reportedly a teacher at Copthorne Church of England Junior School.
Reports suggest the other victim, Sandy Seagrave, 76, had tried to come to the aid of Mrs Appleton.
A 37-year-old man who was arrested on suspicion of murder is in a "very unstable condition" in hospital.
He was found injured inside the property.
Detective Chief Inspector Alex Geldart confirmed it was not a knife attack and said one of the victims was known to the suspect.
A marked police car and uniformed officers were outside the house in Hazel Way on Christmas Eve, as people laid flowers and paid their respects.
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