The Soviet invasion of Hungary was good and based and one of the few correct things Khrushchev did. It's worth bearing in mind the uprising in Hungary coincided with Israel, France and UK's attack on Egypt.

It was a mix of a popular uprising against Khrushchevs faked "secret" speech about Stalin which enabled the fascist elements (paid, armed and trained by US and UK) of Hungarian society to gamble their chance on getting rid of socialist rule.

Fascists marked Communists homes with a white cross and those of jews with a black cross for extermination squads:

The special correspondent of the Yugoslav paper, Politika, (Nov. 13, 1956) describing the events of these days, said that! the homes of Communists were marked with a white cross and those of Jews with a black cross, to serve as signs for the extermination squads. “There is no longer any room for doubt,” said the Yugoslav reporter, “it is an example of classic Hungarian fascism and of White Terror. The information,” continued this writer, “coming from the provinces tells how in certain places Communists were having their eyes put out, their ears cut off, and that they were being killed in the most terrible ways.”

Andre Stil, editor-in-chief of the French Communist newspaper, LHumanite, arrived in Budapest on November 12. He toured the city and conferred with many Communist and other survivors of the days of White Terror. His account is substantially the same as the reports sent in by Times and Tribune and Commonweal and Commentary and U.S. News and Life and Politika eyewitnesses, fascistic mass murder reminding one of the Berlin days of 1933—and the Budapest days of 1919. Thus:

After the tortures, those who were still breathing were hanged Even dead people were hanged. The corpses of those hanged were in such a state that many could not be recognized. The trees in Republic Square still bear the traces. These corpses, in all parts of their bodies, were bored through with bayonet thrusts, assailed by kicks, tom by nails, covered with expectoration...

(Herbert Apheker, The Truth About Hungary, p.220)

CIA sent terrorists to Hungary under the RED SOX program (Horthy here was the leader of the Hungarian fascists under 23 years of fascist rule in Hungary until Soviet liberation).

“The CIA sent RED SOX/RED CAP groups in Budapest into action to join the Freedom Fighters and to help organise them… Radio Free Europe, and the RED SOX/RED CAP groups encouraged the rebels.” Often since denied, this was something known at the time to those in the know. For example, on November 10 1956, the FBI tapped a conversation between Pagie Morris and Jay Lovestone. Morris said “I know the whole thing… Do you remember when I said to you that it was criminal to incite a revolution and a rebellion, and not to follow it through? … Well, the Wisner crowd incited it… And the Horthy crowd has been in it… That Radio Free Europe is the crowd that’s behind it.”

(James Jesus Angleton, the CIA, and the Craft of Counterintelligence, by Michael Howard Holzman, pp. 150-160)

The CIA chief in Vienna recalled that these “were very sad days” – we sat powerless on the sidelines watching the Soviets preparing to crush the revolution.

(ibid)

Weapons were British and American

Some of the weapons used were American, and others almost certainly British. Mr Smith says MI6 and the CIA had buried arms caches in the woods around Prague and Budapest for use by “stay-behind” parties or fifth columnists in case of war.

(MI6 trained rebels to fight Soviets in Hungarian revolt, The Independent)

Hungary, in 1954, was considered a "weak spot" of the Soviet Union according to US committee '

Again on New Years day, 1954, the Foreign Relations Committee of the US Senate released a study, through its chairman, Senator Alexander Wiley which spoke of "accumulating tensions" and mounting "sabotage and underground activities" in Eastern Europe and referred in particular to Hungary as being the most tender spot - the "weakest link

(Truth About Hungary p.112)

The mid-1950s were regarded by the British and the United States as the last chance to challenge Soviet dominion over eastern Europe. The Eisenhower administration had been elected on a platform of “liberating” the Soviet satellite states, but in the 10 years since the Allied victory in Europe, the Soviet Union had strengthened its hold over the central and eastern part of the continent.

USA was planning on WW3 with Soviet Union in 1943 (2 years before WW2 ended) whilst the British - at war with Hungary at this time "looked on at favour of Horthy" (Horthyism was the brand of fascism in Hungary in power for 23 years prior to Soviet liberation which was only more and more influenced by Nazism as the alliance with Austria and Germany deepened during that period and was to be the main fighting force in 1956)

By April 3, 1943, the editors of The Nation, in discussing “Russia After the War,” warned that many of the rich insisted on the inevitability of World War III—a “thought entertained by powerful forces in the United States which fear any modification of property relationships and are made uneasy by the possible existence of a powerful and successful collectivist state in the world.”

Specifically, in terms of Eastern Europe, as Doreen Warriner writes: “In 1944 all the anti-Soviet elements in the Balkan capitals believed that America and Britain would invade the Balkans after the defeat of Germany,” (cited work, p. 21n.).
Leigh White, an American correspondent in the Balkans, writing in 1944, commented upon “the disreputable dynasties (there) of which our Metternichs of the State Department and Foreign Office are apparently so enamored” (cited work, p. 459). The distinguished English historian, Professor A. J. P. Taylor, in his introduction to the Memoirs of Michael Karolyi, declares that: “Even in the Second World War, when Hungary was an enemy state, and democratic Hungarians, one might have thought, our only friends, the British Foreign Office looked with favour on Horthy, Kallay and the rest, while Michael Karolyi was held at arm’s length.”

(Herbert Apheker, The Truth About Hungary, p.71)

Americans gravitated toward the fascist elements in Hungary at the end of WWII

When I left Italy in the Summer of 1945 (writes Mr. Riegel), talk of an inevitable war with Russia was fashionable with the Catholic Right and the small cynics who know the answer to everything. Arriving in Hungary, I found this same inevitability of war an article of general faith, intensified by a heritage of Nazi propaganda and wishful thinking.

He found, in agreement with all other observers—the testimony of some of whom has been offered on earlier pages—that .. fascism and para-fascism, with their off-shoots of anti-Semitism and clerical reaction, are still strong forces in the country.”

These forces gained encouragement from the American officials, for in Mr. Riegel’s words: “The Americans gravitate toward the most dubious elements remaining in Hungary, the remnants of the gentry, industrialists, the higher clergy, and the motley assortment of fascists and opportunists.”

(ibid p. 73)

NATO furnished support to the fascistic elements of the Horthy fascists with:

The Mutual Security Act (of 1951) has as its stated aim, “to maintain the security and promote the foreign policy and provide for the general welfare of the U. S. by furnishing assistance to friendly nations in the interest of international peace and security.” To this was added an amendment, introduced by Representative Charles Kersten (R., Wis.) and approved by the House (and the Senate and signed by President Truman in October) in the following form, appended to the above:

and for any selected persons who are residing in or escapees from the Soviet Union, Poland, Czechoslovakia, Hungary, Rumania, Bulgaria, Albania, Lithuania, Latvia, and Estonia, or the Communist- dominated areas of Germany and Austria, or any other countries absorbed by the Soviet Union, either to form such persons into national elements of the military forces of the North Atlantic Treaty Organization or for other purposes, when it is similarly determined by the President that such assistance is important in the defense of the North Atlantic area and of the security of the United States (Congressional Record, August 17, 1951, vol. 97, p. 10261).

(ibid p.95)

credit to /u/JoeysStainlessSteel