The committee recognized that there may have been a conspiracy  that was work of a group of citizens that might have been unhappy or dissatisfied with the President's policies.

Which individuals or groups within the United States wanted JFK dead? Why did they resort to murder as opposed to working within the democratic system of political change? Is it possible that JFK was too popular to remove through the electoral process, and so his death was the only way available to steer American through their course of history?
Those identified as possible members of the conspiracy to assassinate JFK easily range from those with important implications within America's government, such as the Defense Department and the CIA, to those for who held no major significance in respect to American social or governmental institutions. The final question deals with the role of Lee Harvey Oswald. Did he act alone? Was the shot that actually killed JFK shot from a second gunman on the grassy knoll in front of the president's limousine, while Oswald fired from behind in the Texas School Book depository? Officials believe that there is no possible way a conspiracy could have existed in Dealy plaza because no firm evidence pointing to a second gunman exists.

Even without physical evidence of conspiracy at the scene of an assassination, there would be of course a conspiracy if others assisted Oswald in his efforts. The Warren Commission recognized that a first premise in a finding of conspiracy may be a finding of association. With much respect to Jack Ruby, the Warren commission similarly found no connection between Ruby an Oswald, or between Ruby and others. Based on available evidence, the committee could not preclude that individual members of an pro-Castro group were involved in the assassination.

The House Committee on Assassinations contradicted the Warren Commission's findings on the role of Oswald. To be precise and loyal to the facts, the committee was compelled to find that the idea of a conspiracy could not be discounted because the Warren Commission did not consider all the facts. The Committee criticized the Warren Commission for being overly concerned about public reaction to the possibility of a conspiracy, which determined a different path for the Warren Commission's investigation: 

[O]n the subject that should have received the Commission's most probing analysis--whether Oswald acted in concert with or on behalf of unidentified co-conspirators the Commission's performance, in the view of the committee, was in fact flawed.(13) In its effort to fix responsibility for this failure, the committee, as noted, found one of the primary causes was the absence of the full and proper cooperation of the FBI and the CIA, along with the time pressures and the desire of national leaders to allay public fears of a conspiracy.
 
(Source: http://www.archives.gov/research/jfk/select-committee-report/part-1d.html#wc)

The House committee left open the possibility that President Kennedy was probably killed due to a conspiracy of CIA and organized crime elements.