In the The Club Election (1953), her character is named "Lillian Appleby". In all later episodes, the character's first name was inexplicably changed to "Caroline".
[on playing Caroline Appleby on I Love Lucy (1951)] She was sort of Lucy's nemesis because both had young sons they would brag about. There was a rivalry there, but she had to be nice, Lucy had to be nice to Caroline, because her husband owned the television station.