A second person was also arrested but apparently on an unrelated offence since police said they didn't believe that person was connected to the abduction.
Authorities have said Fletcher, 34, was jogging around 4am on Friday when a man approached her and forced her into an SUV after a brief struggle. Fletcher was reported missing when she did not return home.
Police said a witness reported seeing Abston cleaning the inside of the SUV in question a few hours after the abduction and that he was "behaving oddly."
Fletcher is the granddaughter of the late Joseph Orgill III, a Memphis hardware businessman and philanthropist, according to news outlets. The family has released a video statement asking for help in finding Fletcher and offered a $50,000 reward for information in the case.
"We believe someone knows what happened, and can help," Fletcher's uncle said in the video.