Birth registered as "Katherine (sic) Elizabeth" – almost certainly an error, given that her mother's name is also recorded as "Katherine".
Possibly the same "Catherine E. Blackmore" who:
was the mother of a Catherine Eveline Blackmore, born at Fremantle in 1918, and;
died 1937, at Perth Hospital; last address, Stoneville. (Definitely not Catherine Blackmore, née O'Dea, d. 1949, Bassendean.)
Sources
Private family records;
Registry of BDM Western Australia;
FamilySearch;
Ancestry(.)com
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DNA Connections
It may be possible to confirm family relationships with Catherine by comparing test results with other carriers of her ancestors' mitochondrial DNA.
However, there are no known mtDNA test-takers in her direct maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Catherine: