Communists are NOT in Putin’s government. Nice try: they’re the main opposition party.
A United Russia (ruling coalition): 238 Duma members
Communists: 92
A Just Russia: 64 (Neo-socialists and state pensioners)
LDPR: 56 (Zhirinovsky)
Notice that you do not dispute the other aspects of my post, which, obviously, make a question as to pubic political affiliation irrelevant. The government is largely run by "former" communist party members and KGB officers, including Putin.
As for the communist party being an "opposition party":
“In the first stages of Zyuganov’s creation of the Communist Party of the Russian Federation (not without some participation on my part, as well as Prokhanov…), efforts were made to interpret and conceptually appraise the presence of the national component in the Soviet worldview (National Bolshevism), but this initiative was abandoned by the leadership of the [Communist Party], which had occupied itself with some other matters…. However, on the level of rhetoric and first reactions, Russian Communists in all senses present themselves as confirmed national conservatives – sometimes even as ‘Orthodox Monarchists.’” - Alexander Dugin, The Fourth Political Theory The Russian Orthodox church itself has a patriarch who is a known KGB agent.
What things look like, and what they actually are, is a distinction to learn if you wish to understand Russia.