Both women are from the borough of Queens, the New York City Police Department said.
Authorities say Liverpool is described as emotionally disturbed with a history of schizophrenia and bipolar disorder.
A former home health aide charged with shoving a woman in front of an oncoming subway train under New York's Times Square insists she's not guilty of murder.
Liverpool, 30, was ordered held without bail.
The victim, identified Tuesday as Connie Watton, 49, of Queens, tumbled into the tracks as a southbound No. 1 train approached.
Meanwhile, witnesses said Liverpool-Turner attempted to calmly walk out to the sidewalk, as a throng of police officers filled the station. She was pronounced dead at the scene by police. Despite the confession, it was dismissed after two witnesses told investigators that the deceased woman jumped in front of the train by herself, which prompted a suicide ruling in her death.
She was arrested and, late Monday night, charged with second-degree murder, WCBS reported.
"A passerby followed a suspect up the stairs and pointed her out to a cop, which is what we ask people to do - when you see something, say something", an unnamed police official told DNA Info.
Police also understood Fiedler was undergoing treatment from a therapist for a personal issue before she died, DNAinfo reported.
Police were looking at video surveillance to determine what led to the attack, Manhattan Chief of Detectives William Aubry said.
Mari said Liverpool worked as a home aide up until three weeks ago, has no prior convictions and has lived at the same address for three years.
The newspaper reported that investigators were looking into whether or not Liverpool-Turner had any involvement in an October 19 death at the Union Square subway station.
In 2012, Erika Menendez, a mentally ill woman who had a history of attacking strangers, shoved an immigrant from India off a subway platform in Queens. Kevin Darden pleaded guilty to the push and is awaiting sentencing. The victim, Sunando Sen, was 46.
A fatal subway push in 1999 led to state legislation, called Kendra's Law in honor of victim Kendra Webdale, allowing supervision of certain psychiatric patients outside of institutions to make sure they're taking medications and don't present a public safety threat.