From what I've noticed, white supremacists often have conflicted feelings about Jewish people. On one hand, they're quick to stereotype and ostracize them. On the other, they love using the Jewish community as an example of white people being oppressed based on race (never mind the fact that Jewish people were oppressed for being Jewish, not for being white, and were even considered an ethnic minority as a means to justify the persecution. Also, they were attacked by other white Europeans, not POC). Additionally, racists often assume that all Muslims are anti-Jewish and will cling to that as an Islamophobic stereotype, even if they themselves feel hostile toward Jewish people.

Another factor I've noticed is how they will either say that race isn't the same as nationality/religion or that it is, depending on whether that will further the "white people are oppressed" narrative. This is a pattern that occurs often:

Person: "White people are targeted. Look at what happened to the Irish and the Jews!"

Then the same person will rail against affirmative action, calling it "racism against whites." When told that Irish and Jewish scholarships exist and benefit a lot of white people, they'll backtrack and say, "Well, that doesn't count because those benefits are based on nationality/religion, not race."

Blather, rinse, repeat.

Islamophobia and racism

Many will say that anti-Islamic sentiment is about objection to the religion, rather than about racism, because Islam is a religion and not a race. I used to say the same thing. But while Islam itself is not an ethnicity, it's closely associated with brown-skinned people and therefore Islamophobia involves a lot of racial profiling. Islamophobes tend to have a much bigger issue with race than with the actual beliefs of extremist, fundamentalist Islam. If they genuinely had a problem with the misogyny, homophobia, and justification for violence that exists in the fundamentalist sects of pretty much every religion, then they would have an equal objection to white fundamentalist Christians who hold those same attitudes. And yes, I know that not all Christian fundamentalists are white. Bigots usually excuse the white ones, though, or at least grant them far more leeway.