Yes. But you have to go beyond page one of Google results because there is so much slop, human and otherwise. See [1] for a technical overview. See page 14 of [2]
for the strategic and political aspects of who controlled what in a crisis.
"Cheget No. 1 (the Soviet premier's portable terminal) had
priority status: its owner could enact heightened level of
alert and, after the early warning system registered the
launch of U.S. nuclear weapons, give launch order (transmit
codes unblocking the launch command)." That's the key point. The premier of the USSR did not have personal first strike authority.
Putin, however, has said that he now does. Trump has first-strike authority too.
The definitive book on this seems to be "Russian Strategic Nuclear Forces"[3], an English translation from MIT Press of a Soviet-era document. That's what the good sources cite.
[1] https://subhranil.com/the-dead-hand-system-a-technical-analy...
[2] https://npolicy.org/wp-content/uploads/2021/08/Controlling_S...