I better know them as Bettongs.

http://www.calm.wa.gov.au/projects/west_shield_article.html

The website above also has good information on the threat on small ground nesting or living mammals, marsupials and other native species from the introduced cat and fox predators.

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Woylie or Brush-tailed Bettong Bettongia penicillata ogilbyi (Family Potoroidae)

Features. Males and females are similar. The head-body length is 300-380 mm long; tail length 290-360 mm; body yellowish grey above, paler below, with a black crest on the tail. When startled, a Woylie hops, holding its head low, the tail extended and the back arched.

Distribution. [see map] Once widespread over much of arid Australia, by 1980 Woylies occurred only in three small forest areas in southern Western Australia. Three subspecies have been named. Those in southern Western Australia form the subspecies ogilbyi (b); the south-eastern subspecies penicillata is extinct (a), and little is known of the subspecies tropica (c); aboriginal records, subspecies unknown are shown at (d).

Ecology / Way of Life. Woylies live in open forests and woodlands in which tussock grasses or low woody scrub appears to be typical. They do not drink, but forage from dusk to near dawn for underground fungi, as well as bulbs, seeds, tubers, insects and resin. Bacteria in the large fore-stomach digest and convert fungi to more suitable products for a more balanced diet, and both fungi and bacteria are digested in the posterior part of the stomach before they are absorbed in the intestine.

Woylies spend the day in an elaborate domed nest, which is built of grass or bark. Like other members of the family, Woylies carry nest materials home in the curled tip of the tail. Individuals hold distinct home territories, in which they nest and feed. Females breed continuously, from their first birth at about 170-180 days old, at intervals of about 100 days. The single young lives in the pouch for about 90 days and stays with its mother (Fig. 2) until the next young is born. Woylies live for about 4-6 years.

Interaction with humans/Threats. By 1980, Woylies were classified as Endangered, as they were known only from small forest refuges at Perup, Dryandra and Tutanning in southern Western Australia. Large stands of Poison Bush, which contains 1080 naturally, may have protected Woylies from foxes, one of their major predators. Woylie populations have recovered enough to re-classify them as being at Lower Risk, following extensive poisoning of foxes with 1080, but their survival is dependent on Conservation action, especially control of fox numbers. Woylies were important as food for Aborigines thought the species' range.

http://www.australianmammals.org.au/Species/woylie.htm