As Margulis first suggested, eukaryotes arose as archaea absorbed bacterial endosymbionts which gave rise to the mitochondrion found in all eukaryotic cells. (Archaea and bacteria were long confused as variants of the same life form, as they look and act a lot alike.)
Carl Woese proposed archaea, bacteria, and eukaryotes as domains in 1977 after discovering archaea.
The iconic rooted 3-domains tree of life shows eukaryotes and archaebacteria as separate groups that share a common ancestor to the exclusion of eubacteria. By contrast, the eocyte hypothesis has eukaryotes originating within the archaebacteria, and sharing a common ancestor with a particular group called the Crenarchaeota or eocytes. ~ English evolutionary biologist Cymon Cox et al
In 1984, American evolutionary biologist James Lake and his colleagues proposed that eukaryotes descended from an archaean kingdom called eocytes (aka Crenarchaeota). Follow-on research showed this to be the case.
Bacteria are by far the most diverse domain, with 92 named phyla. There are 26 known archaeal phyla, and 6 eukaryotic supergroups. Classification of eukaryotes is a mess.
The placement of Eukarya relative to Bacteria and Archaea is controversial. ~ Canadian microbiologist Laura Hug et al
There are 3 domains of life: viruses, archaea, and bacteria. Viruses descended from ancient cells which genetically slimmed down to a pathogenic lifestyle: their presence more energetic than molecular. Eukaryotes descended from archaea, in endosymbiotic partnership with bacteria, and so nominally comprise a kingdom, not a domain (if one leaves the current taxonomic stratification intact).
Adjusting yet adhering to familiarity, Spokes refers to 4 eukaryotic kingdoms: plants, protists, fungi, and animals, which are descended from the domain of archaea. Prokaryotes refers to the domains of archaea and bacteria, with eukaryotes herein stated as kingdoms.
Eukaryotic supergroups may be the new fashion for evolutionary biologists (despite remaining unsettled), but the terminology is awkward and not yet publicly well-known: hence, not arguably superior. Further, supergroups break down into conventional phyla. Phylum is the level below kingdom, generally based upon body plan (shades of Linnaeus).