Frederick A. Rogers was the son of the plant collector Reverend William M. Rogers* of Chetnole, Dorset. He was educated at Keble College, University of Oxford, where he graduated as Bachelor of Arts (BA) in 1897 and Master of Arts (MA) in 1904. He came to South Africa in 1899, the year when the Anglo-Boer War (1899-1902) broke out, as acting lay chaplin to the British forces, and earned a medal and clasp. During that year he collected plants along the Midland railway line in the Cape Colony, but the results were poor owing to drought. Upon returning to England he was appointed curate of St John the divine at Kennington, Kent, in 1901 and the next year was ordained as a priest. He joined the South African Church Railway Mission as assistant chaplain in the Diocese of Pretoria in 1905 and was subsequently attached to the Diocese of Bloemfontein (1906) and Northern Rhodesia (now Zambia, 1908). During 1905-1907 he presented hundreds of plants from various localities in the Transvaal Colony, the Orange River Colony (now the Free State) and Natal to the Transvaal Museum, Pretoria. In 1908 he travelled up to the Zambezi River and on his return presented a collection of more than 1000 plant species to the Albany Museum, Grahamstown. These had been collected in the Cape Colony, the Transvaal, and along the railway lines of Zimbabwe, Natal and Mozambique. Many had been identified by the British Museum, others by Harry Bolus* in Cape Town, and many species were new. More plants from Zimbabwe or Zambia were sent to Kew Gardens in England, and to the South African Museum, Cape Town.
After contracting malaria Rogers returned to England in 1909 for a stay of 18 months. During this period his cottage in Bulawayo burned down and a manuscript he was preparing on the history of the mission was destroyed. From 1911 to 1914 he was head of the South African Church Railway Mission and twice visited Elisabethville (now Lubumbashi), capital of Katanga Province, Democratic Republic of the Congo. During 1911-1913 he presented more plants from Zimbabwe, Transvaal and the Cape Colony to both the Albany Museum and the South African Museum. Another collection of over 2000 plants was presented to the Rhodesia Museum, Bulawayo, in 1912 and formed the nucleus of the Rogers Herbarium. He first collected on the Witwatersrand in 1915, and in 1918 collected many plants there with Professor Charles E. Moss*.
Rogers not only collected extensively all over southern Africa, but also received plants from other collectors, including George Thorncroft* of Barberton. His more than 24 000 specimens are housed in various local and overseas herbaria, namely those of the British Natural History Museum; Kew Gardens; the Conservatoire et Jardin botaniques, Geneva; the National History Museum of Zimbabwe, Bulawayo; the Albany Museum, Grahamstown; the Moss Herbarium, University of the Witwatersrand; the National Herbarium, Pretoria; and the Compton Herbarium, Cape Town. A number of new species from his collection were described by S. Moore in the Journal of Botany during 1913-1921. Several species were named after him by Moore (Anisotes rogersii, Thespesia rogersii, Oldenlandia rogersii), by J. Burtt Davy* (Albizia, rogersii), and others. Rogers's only botanical publication was a Provisional list of flowering plants and ferns found in the divisions of Albany and Bathurst, Cape Colony (Grahamstown, 1909, 21p), based on earlier works by W.H. Harvey* and S. Schönland*.
By 1906 Rogers was a member of the South African Association for the Advancement of Science. Though his membership later lapsed, he rejoined the association in 1918 and was elected a member of its council for 1918/9. By 1917 he was a member also of the Rhodesia Scientific Association. From 1914 to 1921 he served as archdeacon of Pietersburg (now Polokwane). In 1922 he became chief secretary of the Society for the Propagation of the Gospel in Foreign Parts, but during 1923-1924 was also general treasurer of the Diocese of Grahamstown. He was assistant secretary of the Missionary Council of the Church Assembly from 1925 to 1928, and chief chaplin at Baghdad, Iraq, from 1928 to 1929. Later he settled in East Grinstead, Sussex.