I don't believe Accarodo or Auippa had any "hands-on" involvement with the murder. I firmly believe each played a part relative to giving the ok. None of these top echelon guys would do the actual killing because they had others do it for them. The general feeling is that someone very close to Giancsna was responsible for the murder.
The .22 caliber semi-automatic was found some distance from Giancana's home on a route that the suspected killer would have taken on his way home. I believe the gun was traced to some place in Florida.
LaPietra good very well have been responsible for disposing of the gun, but there is simply no way to know for sure. All the "players" are dead and all we're left with are inferences and opinions, Including mine.