Short, Magazine Lee-Enfield Mark III Star Bar is the full designation.
With that somewhat wonky Crown on the right side of the Butt Socket, it is definitely produced by Standard Small Arms or by its Government successor, National Rifle Factory. They were the same plant, just under either private or Government ownership. SSA rifles can be found dated 1916, 1917, (rarely) 1915 and sometimes 1918. There was considerable mix-up in dating of Bodies, with many SSA-manufactured Bodies being completed and out the door with NRF markings, thus demonstrating the great efficiency of Government management.
The marking will be either SSA or NRF on this one. The marking sometimes is not stamped very well, but it will be found on the action itself, just to the shooter's left of the actual bolt raceway: not the best position for a factory marking, being more than a bit exposed to damage. But it WILL be there.
Looking closely, this one appears as if it may not have been cut for a Cutoff, in which case it would have been a Mark III* (Mark three STAR) originally. The STAR would have been BARRED out when a Cutoff was fitted, later, likely post-War. By rights, it should have become a STAR BAR STAR when the Cutoffs were removed permanently just before War Two! But in fact WAS IT? Could this be an original remarked Mark III by SSA which NRF completed as a III*? Only close examination can tell.
It can be really confusing at times; I have a Star Bar Star here with an 1897 Body and a 1943 barrel! War Two was very much a come-as-you-are party in its first year or so and everything which went BANG! or could be made to go BANG! was dragged out and rebuilt. Fortunately, wiser heads did sometimes prevail and by 1942 things generally made sense.