This William Whitmore is the nephew of our William Bernard Whitmore. He originally joined the Royal Warwicks, but as you can see from the address here, he was transferred and served in the Machine Gun Company of the 73rd Infantry Brigade of the 24th Division (on 5th March 1918 this became the 24th Machine Gun Battalion). He arrived in France (Le Havre) on 11th March 1915 and remained for the duration of the war.

The battalions in the 57th Brigade were:

The 24th Division was involved in the following action: Please note: as with all tables like this one, remember that it lists only engagements with the enemy. Troops would spend considerable amounts of time each year in the trenches, and within firing range of the enemy. Shelling and sniping would be a permanent danger, which lessened (but did not disappear) when there was no major engagement.

You'll notice that William Bernard was also engaged in the Battles of Guillemont and Vimy, and the pursuit to the Selle. However, in each of these engagements, the two Williams were assigned to different Corps.

Campaign Battle Dates Present
  Battle of Loos 25 Sept. - 8 Oct. 1915
Battle of the Somme Battle of Delville Wood 15 July - 3 Sept. 1916
Battle of Guillemont 3 Sept. - 6 Sept. 1916
The Arras Offensive Battle of Vimy 9 Apr. - 14 Apr. 1917
The Flanders Offensive Battle of Messines 7 June - 14 June 1917
3rd battle of Ypres Battle of Pilckem 31 July - 2 Aug. 1917
Battle of Langemarck 16 Aug. - 18 Aug. 1917
2nd battle of the Somme Battle of St Quentin 21 Mar. - 23 Mar. 1918
Actions at the Somme Crossings 24 Mar. - 25 Mar. 1918
Battle of Rosieres 26 Mar. - 27 Mar. 1918
Battle of the Avre 4 Apr. 1918
Battle of the Ancre 5 Apr. 1918
Battles of the Hindenburg Line Battle of Cambrai 8 Oct. - 9 Oct. 1918
  Pursuit to the Selle 9 Oct. - 12 Oct. 1918
Final advance in Picardy Battle of Valenciennes 1 Nov. - 2 Nov. 1918
Battle of the Sambre 4 Nov. 1918
Passage of the Grand Honelle 5 Nov. - 7 Nov. 1918

Taken from The British Army in the Great War, and The Regimental Warpath.