Details of his training are unrecorded. However, Groom was appointed a flying officer on 30 January 1918 and placed on the General List of the Royal Flying Corps. On 18 March, he was assigned to No. 20 Squadron as a Bristol F.2 Fighter pilot. On his first combat flight, his formation leader fired a Very flare that landed in the rear cockpit of Groom's Bristol. While the observer burned his hands smothering the fire that threatened to set off munitions, Groom safely landed. Having survived friendly fire, Groom began to triumph over enemy fire on 8 May 1918, when he began a string of eight victories that would take him through to 30 July. Groom's final tally was three enemy planes set afire, four otherwise destroyed, and one driven down out of control; his observer/gunner for all these victories was Ernest Hardcastle. Groom went on leave, was laid low by influenza, and did not return to combat in France. Instead, upon recovery, he was posted to No. 111 Squadron in Egypt.
Between the World Wars
On 1 August 1919, Groom was granted a permanent commission as a lieutenant. On 15 December, he switched to No. 14 Squadron in Palestine and on 22 May 1920, he continued Middle Eastern service with a new posting, to No. 55 Squadron. On 25 March 1922, Groom returned home to begin a decade of staff assignments, enlivened only by being promoted from flying officer to flight lieutenant on 1 July 1924. He was given command of No. 28 Squadron on 26 October 1932. On 1 October 1934 he was promoted to squadron leader, his rank catching up with his position. He moved on to command of No. 58 Squadron, then on to staff work at Headquarters Bomber Command, even as he continued to ascend in rank. On 1 January 1938, having earned an OBE at Bomber Command, he was promoted to wing commander.
On 1 September 1940, he was promoted to temporary group captain and became Station Commander at RAF Marham. In 1941, he moved back into staff work in the Directorate of Plans. The following year, he became head of the RAF Air Staff planning for Operation Overlord. As part of his duties there, he accompanied Air Chief MarshalTrafford Leigh-Mallory to North Africa to study operations there and bring home lessons learned. In the meantime, he continued his climb through the ranks: on 18 November 1942, he was confirmed as a group captain. On 1 June 1943, he was bumped up to temporary air commodore and on 8 September 1943 he was promoted to acting air vice marshal on appointment as Senior Air Staff Officer at Headquarters Second Tactical Air Force. On 8 September 1944 he was promoted to air commodore while acting as air vice marshal. In August 1945 he became Air Officer Administration at Headquarters Flying Training Command.