© the copyright holder. Photo credit: Manchester Art Gallery
The Mapping Sculpture entry for this sculptor, which mentions this specific bust, spells her surname Aarens, not Aaerens as in the Art UK entry. It also says she exhibited in Liverpool in 1903, which means she was active at least 1903-1934. There seems to be very little known about her at present.
Gallery correspondence with the artist confirms the spelling of 'Aaerens'. It is also known that Betty Aaerens exhibited in 1932 with the Lancashire and Cheshire Artists' Exhibition in October - November 1932, with two artworks: 226 Tanya and 235 Portrait of a Youth. Correspondence between the Gallery and Mrs Esther Barker, wife of the 'Noel' portrayed in the Manchester Art Gallery bust, suggests Aaerens left Manchester in 1934 to live in London, where she later married.
The 1903 Liverpool exhibition was almost certainly the Liverpool Autumn Exhibition, held annually from 1873 to 1938 (apart from during WWI), and from 1877 at the Walker Art Gallery in Liverpool The Walker is presently in the process of digitising the first 10 years of the catalogues (1873-1882), but the rest of the series can only be accessed in hard copy. Copies of the catalogues are kept in the curatorial offices of the Walker and in the National Art Library at the Victoria & Albert Museum, London, but neither offices nor NAL are open to the public during the pandemic. Even if the 1903 catalogue was accessible it would only provide the address of Betty Aaerens.
If she had already been active for some 30 years by 1934 and then left for London and got married, that sounds rather late for such a move, though it's possible.
For her submissions to the Royal Academy in 1904, 1905, and 1907, 46 Quarrendon Street, Parson's Green, London (one of the Mapping Sculpture addresses for Aaerens), was the address of the sculptor and medalist Margaret Winser (sic), who died a spinster at Ratsberry, Tenterden, Kent, in 1944. She had lived there from 1909 until her death. In 1929, she submitted a bust of "The late Dame Ellen Terry, G.B.E.". Quarrendon Street might have been an artists' house or perhaps there is a family connection.