Hmmm... As to women, you're talking about the religious evangelicals. That faction is their own distinct kettle of fish.
No, unfortunately for moderate Republicans they are not only in the same kettle, they are running the kettle. Apparently a large majority of Congressional Republicans and apparently about half of the Supreme Court conservatives is of that "faction".
Cite something they did. I dare you.
I'm not going to bother citing since it's so trivial to search, but you can look up a whole bunch of voting regulations intended to suppress minority voting while pretending to address a completely non-existent "voter fraud issue". And there have been leaked political memos, etc talking about how it will reduce minority turn out in some districts, so it's pointless to try to argue otherwise.
Also, opposing affirmative action in education. Clearly there is a debate on the topic, but there is no question the intent was to give a boost to a population which had been denied the same opportunities white males have had in college education. I think everyone hopes that will become unnecessary in the future but a couple of centuries of social and financial discrimination can't just be fixed in a couple decades.
Also, he LGBT community is clearly in the same position African Americans were in the 50s-60s, and being treated just as badly by conservatives. It's pretty clear once the old ones die off the more tolerant next generation will be more accepting, though. It's already happening, luckily, and my guess is gay rights will be a non-issue in 10-20 years. Doesn't mean Republicans haven't been fighting it as much as they can...
And as for both minorities and immigrants - 221 members of the House Republicans voted to defund the DREAM Act and work to deport aliens who were babies when brought into to the US illegally, and often didn't even know they were not citizens. Crack down on illegal immigration when it happens, I agree, but don't kick people out who didn't even know they were illegal and have been here since before they could speak any language, let alone their *native* English...