Concept:
Archaebacteria are unique prokaryotic organisms that differ significantly from eubacteria in the composition of their cell membrane and cell wall. One of their most distinctive features is the presence of branched-chain lipids in their plasma membrane.
Unlike bacteria and eukaryotes, which possess unbranched fatty acids linked to glycerol through ester bonds, archaebacteria possess branched hydrocarbons attached to glycerol through ether linkages. These adaptations help them survive in extreme environments such as high temperature, high salinity, and highly acidic habitats.
Step 1: Recall the membrane composition of archaebacteria.
The cell membrane of archaebacteria contains:
\[
\text{Branched-chain lipids}
\]
and
\[
\text{Ether linkages}
\]
instead of the ester-linked unbranched fatty acids found in other organisms.
Step 2: Analyze the given options.
Actinomycetes possess ordinary bacterial membranes.
Mycoplasma lacks a cell wall but does not contain branched-chain archaeal lipids.
Eubacteria possess ester-linked membrane lipids.
Archaebacteria possess branched-chain membrane lipids.
Step 3: Select the correct organism.
Therefore,
\[
\boxed{\text{Archaebacteria}}
\]
contain branched-chain lipids in their cell membrane.