“Why would those who conducted the investigation, or held onto it all these decades be embarrassed?”
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Because they knew Oswald was a loose cannon and a threat and they screwed up by not preventing him from murdering JFK.
The whole thing seemed like a setup to me. Right down to the fact they allowed the assassin to be assassinated. I say allow, because that's exactly what they did.
Btw, I'm not sure what group or who you think did it or had it done, so I'm just sayin.☻