According to the Qur'an, God has absolute power over all things.
Satan, as a created being, is under God's power and only has some capability to influence or affect people because God has allowed this for the duration of this world or universe.
God has given Satan free will, just as God has given free will to human beings and other jinn. (According to most Muslim exegetes, Satan is one of the jinn, not an angel or fallen angel.) However, this free will is not absolute: while we are responsible for our decisions in the hereafter, and lead ourselves to heaven or hell, we cannot always do everything we want since sometimes the divine decree/destiny affects what happens in our lives. So, we are still under the power of God, even if we have free will in how we respond to matters that are fated.
Therefore, Satan is not an equal contender to God and there is no cosmic battle between good and evil, even if sometimes an individual person may go back and forth towards or against Satan.
(However, the idea of a cosmic battle between good and evil predates Islam, and has hence influenced some cultures, and also seems to be part of the human psyche, so you will sometimes hear Muslims framing things like that. But that is not theologically correct according to the Qur'an.)
Some scriptural references from the Qur'an:
* Satan has no powerful over the faithful who seek refuge with their Lord (16:22).
* On Judgment Day, Satan will tell people, "God made promises that were true, and I made promises to you that I did not keep. I had no authority over you, except that I called you and you obeyed me. So do not blame me; blame yourselves. I cannot aid you now, and you cannot aid me now. I disassociate myself from the way you associated partners with God." (14:22)
* The story of when Adam is created, God orders Satan to bow to him. Satan refuses, and God expels him from the celestial realms. Satan swears to misguide humanity, so God allows him to do that for the duration of this world or universe, after which time Satan and those who choose to obey him will enter hell. (7:11-7:18, and elsewhere in the Qur'an)
* Satan's power is to whisper and delude. (4:120)
* Satan's plots are weak. (4:76)
* People choose by their own free will to obey (and hence worship, since obedience is considered a form of worship) Satan and are held accountable for it. (36:59-63)
These sections of the Qur'an can be read further at quran.al-islam.org or other websites if one does not have a printed copy of the Qur'an.
However, insofar as Satan's power in this world among humanity is proportional to how many people choose to support, follow, or obey him, one could say that, to some degree, Satan's power depends on people's choices. This is similar to a country which has a president who has full authority to exert his/her will (for instance, a military and police force which will do what he says), but where the people follow what a social influencer says. He may allow this and be the ultimate authority, and could shut down the influencer if he wanted, but, in practice, the social media influencer is asserting some authority due to the people's choice.
So, from this angle, one could say that sometimes Satan's influence is greater or lesser, depending on where one is, what era one is living in, etc. Certainly there has been a lot of evil in the world in the 20th century, however, it is also quite possible that Satan was not behind this and is simply standing there smiling and laughing as human beings make a mess of things themselves.
Some of this is apparently also due to human choice, especially the move away from the divine and the sacred in some heavily secular ideologies; the Qur'an says: "They forgot God, so God forgot them". (8:67) People can do this with or without Satan.
Still, the Qur'an says: Had it not been for the grace and mercy of God Allah upon you, you would have followed Satan, except a few of you (4:83). So this reiterates the idea that God is the ultimate authority, not Satan.
I also like to think that, despite the evils of the world, most people are ultimately good at heart and want for things to go aright; we are just going through a difficult time.
Hope that helps!
