Who were the 'synagogue of Satan'? In the letter to the Church of Philadelphia in Revelation 3, John writes concerning the synagogue of Satan and those who claim to be Jews, but are not.


  Behold, I will make those of the synagogue of Satan who say that they are Jews and are not, but lie—behold, I will make them come and bow down before your feet, and they will learn that I have loved you. (Revelation 3:9 ESV)


What group was it that claimed to be Jews and were not?  What was it about them that made them the synagogue of Satan?
I would guess that the reference to the synagogue of Satan is not to some local synagogue of Satanists that were pretending to be Jews, but instead to all those everywhere who claimed to be Jews but were not really, spiritually, Jews.  Consider Jesus' words in John 8:


  They answered him, “Abraham is our father.” Jesus said to them, “If you were Abraham's children, you would be doing the works Abraham did, but now you seek to kill me, a man who has told you the truth that I heard from God. This is not what Abraham did. ... You are of your father the devil, and your will is to do your father's desires. John 8:39,40,44


In other words: "your actions show you aren't Abraham's children (i.e. you aren't Jews); you're children of Satan."

And then, also consider this passage, which divides everyone into two groups: children of God and children of the devil:


  By this it is evident who are the children of God, and who are the children of the devil: whoever does not practice righteousness is not of God, nor is the one who does not love his brother. 1 John 3:10


Also, consider Galatians 3:7-10:


  Know then that it is those of faith who are the sons of Abraham. And the Scripture, foreseeing that God would justify the Gentiles by faith, preached the gospel beforehand to Abraham, saying, “In you shall all the nations be blessed.” So then, those who are of faith are blessed along with Abraham, the man of faith."


Finally, Romans 9:6-8:


  ... not all who are descended from Israel belong to Israel, and not all are children of Abraham because they are his offspring, but “Through Isaac shall your offspring be named.” This means that it is not the children of the flesh who are the children of God, but the children of the promise are counted as offspring.


So we see that the true sons of Abraham, the true Jews, are those who are "of faith".  Thus, I would conclude that the passage you quoted is simply referring to Jews who rejected Jesus as Messiah.