Who is The Synagogue of Satan?

What is the "Synagogue of Satan" mentioned in Revelation 2:9 and 3:9.
Both passage identify this group as "those who say they are Jews but
lie".

An interpretation of the above texts have surfaced which interprets
these texts to refer to those Jews who do not accept Yeshua as the
Messiah. But an honest look at the Scriptures will show that it is
not possible to identify the "Synagogue of Satan" in this way.

To begin with it is important to recognize that the same "Yochanan"
("John") wrote both Revelation and the Gospel of Yochanan. Not only
is this the traditional understanding, but there are a number of
common elements that point to the common authorship of these two
books. Both books identify the Messiah as the "lamb" (Jn. 1:29; Rev.
5:6, 8, 12; 14:1) and as the incarnate "word" (Jn. 1:1-3, 14; Rev.
19:13) and both refer to the "living waters" (Jn. 4:10; 7:38; Rev. 22:1)*.

Now we must look at how Yochanan uses the word "Jew" in his own
writings. Yochanan himself frequently invokes the term "Jews" to
refer to Jewish people who rejected Yeshua as the Messiah. For example:

"therefore did the Jews persecute Yeshua" (Jn. 5:16)
"the Jews sought more to kill him" (Jn. 5:18)
"the Jews then murmured at him" (Jn. 6:41)
"the Jews sought to kill him" (Jn. 7:1)
etc. etc. etc.

CLEARLY Yochanan has absolutely no objection whatsoever to using the
term "Jews" to refer either to Jews who accepted Messiah or those that
rejected him. Therefore the statement in Rev. 2:9 and 3:9 cannot be
criticizing this usage of the word "Jews". Otherwise Yochanan would
here be criticizing his own inspired writing in the Gospel of John in
which he does just that himself frequently!

So what does Yochanan refer to here? I believe he refers to the newly
hatched apostate teaching of "replacement theology".