Apparently they did. Or an edited version of it. Max Holland mentions it in his piece linked at the end of this (it's a good read).
From the piece:
Background: The evidence (for me) is that right after the assassination and the arrest of Oswald, the KGB hurriedly put together a report on their involvement with Oswald when he lived there (Norman Mailer goes over this in some detail in his book "Oswald's Tale"). Vladimir Semichastny, head of the KGB, then presented that report to the Central Committee of the CPSU, specifically Anastas Mikoyan. Mikoyan was a top Soviet official (how he survived the Stalin era is a story in itself) and then deputy chairman of the above committee.
On November 25, Mikoyan gave a edited copy of that report to State Department officials in Moscow.
Later, the Warren Commission was given access to the report. Or a summary of it. It's not clear whether it was the report Mikoyan gave or a summary of it. However, the Commission did not include or mention it in its report, e.g., it's not among the Commission's exhibits.
Oleg Nechiporenko, one of the three KGB agents who met Oswald at the Soviet Embassy in Mexico City, quoted part of that report/memo in his book "Passport to Assassination."
The Holland piece in its entirety can be read here (it starts on page 12):
https://www.wilsoncenter.org/sites/default/files/media/documents/publication/f-research_notes.pdf
