Perryman appeared in "The Blues Brothers," "Boys Don't Cry" and "When Zachary Beaver Came to Town."
Police charged 26-year-old Seth Christopher Tatum with capital murder, plus aggravated assault with a deadly weapon in another attack.
Sgt. Joseph Chacon says Tatum showed up at a jail facility Thursday and said he was "pretty sure" he had killed the owner of a vehicle he had stolen. Police later found Perryman's body at a home.
Investigators say they do not believe Tatum and Perryman knew each other.
A jail official didn't know whether Tatum had an attorney.
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