Looking at the history of Islam, it has always used the method of acculturation to infiltrate it's Arabian, African, and Asian neighbors. This is the same game being repeated, over and over...
...We have a problem on our hands.
There's an aspect of Islam that separates it from the other Abrahamic religions... and it's one that people rarely talk about (amid the bickering of "whose book technically has the most violent verse in it", "who did the most recent thing that could be construed as terrorism", etc...)
And that is, Islam (via the texts) also act as a political prescription.
While the other 2 certainly have verses that could potentially influence how a person may vote, that's different from being the aforementioned political prescription.
Early Islamic texts (Quran and Hadith) contain structures and policy guidelines on inheritance, contract enforcement, taxation, warfare, judicial procedure, wealth distribution, and governance.
Islam, compared to other religions, integrated religion and political authority right out of the gate, whereas others developed politically oriented systems later (e.g., canon law).
There's also aspects of Islam that encourage pro-natalist (have a bunch of babies) specifically for the purpose of "outbreeding" the other religions.
There was an Islamic leader (and I believe it was someone from the Muslim Brotherhood, the group you mentioned) who recently did an interview and said - paraphrasing - "for the vision of Sharia's future in Western countries, ballots are the new bullets, we used to say democracy was haram, but now I say participation in democracies using greater numbers will be our jihad"
It's the reason why the only Theocracies left on the globe are all for that that particular religion. (with the exception of Vatican City which I guess technically would count if you wanted to that a real country).