Grover was commissioned into the King"s Shropshire Light Infantry in 1914. He was awarded the for his service in World War I having been wounded three times during the four years of War. He became Commanding Officer of the 1st Bn King"s Shropshire Light Infantry in 1938.
He served in World World War II as Commander of the 11th Infantry Brigade from 1940 and as General Officer Commanding 2nd Division in Burma from 1941.
He devised and implemented a strategy to remove the Japanese from the Kohima region of India leading to the defeat of the Japanese at the Battle of Kohima between April and June 1944. Despite being victorious in this battle, he was removed from command on 4 July 1944 by Lieutenant General Montagu Stopford who was unhappy with Grover"s methods.
Grover went on to be Director of Army Welfare Services at the War Office in 1944 before retiring in 1948. He became Colonel of The King"s Shropshire Light Infantry in 1947.
In 2014, on the 70th anniversary of the Battle of Kohima, a memorial to General Grover was unveiled at Jotsama on the site of his 2nd Division Headquarters.