??
In what way was Michael wrong about Frankie? He knew right away that Roth, not Frankie, was behind the Tahoe shooting.

Michael ruthlessly manipulated Frankie. He scared the s**t out of him by visiting him unannounced. Then, when he told Frankie that Roth was the culprit, Frankie practically passed out with relief. So, when Michael asked him to "help me take my revenge," Frankie said: "Michael, anything..." That's when Michael asked him to stick his head in the lion's mouth by meeting with the Rosatos to "settle these problems." Michael couldn't lose: If they settled, it was one more worry off his mind. If the Rosatos killed Frankie, it'd be further confirmation of Roth being behind the Tahoe shooting. It was Michael at his most cunning--and most reprehensible.
That was the real injustice done to Frankie.

But, injustice or not, Frankie broke omerta and ratted out Michael to the Feds. He intended to bring Michael down with his affidavit and his testimony. He knew the risk he was taking. According to the code he swore to uphold when he was made, he had to die.