I can only assume they were the first to be blamed and I doubt they wanted to be blamed so I was wondering what they did in reaction.
Amazingly, Soviet intelligence believed that Kennedy had been the victim of a right-wing conspiracy, and that Oswald was just the fall-guy — and more to the point, that he was specifically chosen to divert attention from the real plotters and towards the communists (and therefore, towards Moscow.)
The Soviets were quick to confirm privately that Oswald was not, and had never been a Soviet asset — and in fact, they had assumed for months that he was an FBI agent provocateur. According to Christopher Andrew, in The Sword and the Shield:
[Oswald] wrote to the [Communist Party of the USA] in August 1963 asking whether it might be better for him to continue the fight against 'anti-progressive forces' as a member of the 'underground' rather than as an open supporter of 'Communist ideals.' Jack Childs (codenamed MARAT), an undeclared member of the CPUSA who acted as one of its main points of contact with the KGB, warned Moscow that Oswald's letter 'was viewed as an FBI provocation'. The fact that, unknown to the KGB, Childs was himself an FBI agent renders his warning unusually ironic.
The KGB investigation reached a somewhat bizarre conclusion, based on a mixture of ideology and highly circumstantial evidence. This is an extract from the report made to the Central Committee by Vladimir Semichastny, the chairman of the KGB in December 1963, just weeks after the assassination:
A reliable source of the Polish friends [the Polish intelligence service], an American entrepreneur and owner of a number of firms closely connected to the petroleum circles of the South, reported in late November that the real instigators of this criminal deed were three leading oil magnates from the South of the US—Richardson, Murchison and Hunt, all owners of major petroleum reserves in the southern states who have long been connected to pro-fascist and racist organisations in the South.
According to Andrew, the KGB First Chief Directorate's Service A — who specialised in 'active measures' — "may well have been genuinely persuaded that Kennedy was the victim of a right-wing conspiracy", and that they launched a significant propaganda and political warfare campaign designed to attack the 'lone-wolf' conclusions of the Warren Commission and foment the view that there was a domestic conspiracy against Kennedy.
Khrushchev and Kennedy had a surprisingly cordial (if frequently strained) personal relationship; Khrushchev's biographer William Taubman writes:
In the hours following the assassination Khrushchev seemed in a state of shock. Troyanovsky [his foreign policy aide] could see that he took the news as "a personal blow." The next day at Spaso House, where Khrushchev went to sign an official condolence book, he appeared to have been weeping. Besides an official letter of sympathy, he added a personal note to the president's widow.
Both Taubman and Andrew conclude that Semichastny's claims of a right-wing conspiracy persuaded Khrushchev. Andrew writes that "Khrushchev seems to have been convinced by the KGB view that the aim of the right-wing conspirators behind Kennedy's assassination was to intensify the Cold War and 'strengthen the reactionary and aggressive elements of American foreign policy.'"
This belief in a right-wing Texas made Khrushchev and the Soviet diplomatic/intelligence corps deeply suspicious of Lyndon B. Johnson (in part due to his Texas roots), who was viewed in Moscow as a conservative and a hawk. According to Taubman, "Khrushchev was prepared to take risks with Kennedy as a partner, he told his son, but with Johnson in power, 'everything will be different.'"