I don't know what religious groups you have in mind; it's hard to answer in the abstract. In general, the issue that seems to have determined whether a religious group might or might accept eugenics is their stance on contraception, because most approaches to eugenics are directly concerned with the regulation of pregnancy (through various means, some voluntary and some not). So Catholicism in particular was always the most consistent opponent to eugenics for this reason.

In general, ascribing to eugenic ideas and belief in evolution are pretty closely historically linked, but they do not necessarily have to be. Anti-evolutionists still recognize that you can get major differences over generations within species; they are not anti-genetics. Their point of contention is whether species are "fixed," not whether there is lots of room for change within a species. So they recognize that selective breeding can get you dogs that vary in size from Great Danes to Teacup Chihuahuas, but that these are all the same species. The evolutionist would argue: yes, but if you kept that population of Great Danes and that population of Teacup Chihuahuas genetically isolated for a very long time, they would eventually "drift" enough that they would no longer be the same species. The Creationist would say, "nuh uh." Anyway, my point is that eugenics is about artificial selection within a species, not about evolution per se — it is not about making some version of human that is genetically incompatible with the rest of the species. So eugenics and Creationism need not be, in this way, incompatible, however historically linked they are.

It is hard to generalize about all approaches to eugenics just as it is hard to generalize about "religious groups" generally. There have historically been many, many different forms of eugenics. It is one of the reasons that as a basic idea it was so popular among groups that otherwise shared very little political agreement in the early 20th century, though their motivations and methods were quite different. So there were the forms of eugenics we tend to associate with "conservative" approaches, like forcibly sterilizing the mentally ill and disabled, but there were also "liberal" approaches to eugenics, like providing birth control to the poor for them to use voluntarily. To make sense of why a group advocated for eugenics, one has to first look closely at what they mean by "eugenics," and then look at what that advocacy did for their own agenda.

Scientific racism is its own other bucket of worms, again with historical linkages to eugenics but not necessarily intertwined with it. (There were eugenicists who were racists, and there were eugenicists who were not racists, and there were racists who were not eugenicists.) The earliest forms of scientific racism were in fact Creationist and predated evolutionary theories — the work of Louis Agassiz, for example, which argued that Genesis was not a literal account of creation, and that natural history had revealed that many instances of "special creation" had been undertaken over very long periods of time (hence extinctions and Ice Ages and all that), and that it was right to assume that the different races of humans were also specially created, and endowed with different properties just as different types of animals were, and some were meant to rule and others to be ruled, and so on.

This is in fact part of the context of Darwin's own work and interest in evolution (he was violently opposed to slavery and anything used to justify it, which Agassiz's work was), and his second book on evolution, The Descent of Man, is about 50% about the evolution of human races, with the partial goal of arguing that all humans are of the same species and their differences are pretty superficial. Ironically, arguing that all humans descended from a single common ancestor was considered a not-very-scientific argument in the context of Victorian anthropology, because it sounded a lot like the Adam/Eve situation.

Anyway, all of this is just to say that scientific racism is historically more decoupled from evolution than most people realize — it predates it considerably, and is compatible with Creationism as well, even pretty literal readings of Genesis (the Bible does not in any way say directly that Adam/Eve were the only people God created, and having separate instances of creation for other humans actually smoothes over some of the weirdnesses created by the fact that suddenly other people come into the story who are not obviously descended from Adam and Eve — like Cain's wife, who either was his sister, or someone created entirely separately). One should keep in mind that scientific racism is sort of a justifying gloss that sits on top of other, more deeply-held racist beliefs, and like all pseudoscientific justifications is as pliable as the person who wants to justify themselves wants it to be.

Anyway, that is not much of answer except to emphasize the mutability of these ideas (which is part of the reason they were so popular, and have been so hard to extinguish despite their horrors and failures). But if you are interested in both pre- and post-Darwinian debates about scientific racism, George Stocking's Victorian Anthropology is the best book on the subject I know, and quite interesting.