The relationship between the Orthodox Church and the Russian government goes back to the Byzantine empire and the Tsars. It was suppressed (somewhat) during the Soviet regime. Putin has had many photo-ops with the Patriarch in order to show improved relationship between church and state.
I have embedded a video of their protest below and you can judge for yourselves the appropriateness of that kind of protest, in that venue.
Artists such as Madonna and Paul McCartney have offered support the band and see this as an infringement on religious freedom. The group does not have popular support in Russia. Most Russians see their protest in that sacred space as inappropriate.
My own view is that the protest in that venue was ill advised. It had the opposite symbolic impact (in Russia) that the band had hoped for. Culturally, in Russia, protest has been conducted far more subtly. In his novel, Anna Karenina, Tolstoy did not full out oppose divorce laws as they stood in Russia at the time. One could be left with that impression after reading the novel but it was not foregrounded.
Also, Dostoevsky was extremely distrustful of the West and symbolically expressed it through the very prophetic, Grand Inquisitor. Dostoevsky, though, was deeply religious.
The point is that they are at odds with Russian sensibilities, history and culture and they way in which art has been mobilized politically. This is why they do not have support in Russia.
That said, the young women themselves have demonstrated a great deal of depth in their public statements. Here is part of the statement of one of the band members, Yekaterina Samutsevich, at the close of the trial.
Our sudden musical appearance in the Cathedral of Christ the Savior with the song “Mother of God, Drive Putin Out” violated the integrity of the media image that the authorities had spent such a long time generating and maintaining, and revealed its falsity. In our performance we dared, without the Patriarch’s blessing, to unite the visual imagery of Orthodox culture and that of protest culture, thus suggesting to smart people that Orthodox culture belongs not only to the Russian Orthodox Church, the Patriarch and Putin, that it could also ally itself with civic rebellion and the spirit of protest in Russia.
Perhaps the unpleasant, far-reaching effect from our media intrusion into the cathedral was a surprise to the authorities themselves. At first, they tried to present our performance as a prank pulled by heartless, militant atheists. This was a serious blunder on their part, because by then we were already known as an anti-Putin feminist punk band that carried out their media assaults on the country’s major political symbols.
And in a response to a sermon from the Patriarch, the band said (in part) this:
Your Holiness, Patriarch-
A fervent and sincere prayer can never be a mockery, no matter in what form it occurs, therefore it cannot be said that we jeered at, or mocked, the shrine.
We are plagued by the thought that the very shrine, which you consider so defiled, is inseparably linked to Putin, who in very words, brought it back to the Church. And because the of our prayer, asking our Holy Mother to drive out those who defile the brightest ideals of human life in Russia and all possible precepts of the Orthodox Faith, you are perceived as a mockery of the sacred.
In prayer it is evoked that , as millions of Christians were seriously grieved that you allowed the Church to become a weapon in a dirty campaign of dirty intrigues, urging the faithful to vote for a man whose crimes are infinitely far from God's Truth. We simply cannot believe the representative of the Heavenly Father if he acts contrary to the values for which Christ was crucified on the cross. As said by Pushkin, “ It is impossible to pray for King Herod; the Mother of God forbids it.”