Simmons Hardware Company used the "SH Co" logo .
Crescent Fire Arms used various names for "house brand" shotguns and this may be one of them. They used other companies to make shotguns for them. Hartford Arms actually made the guns for both Simmons and Shapleigh Hardware. (Shapleigh bought out Simmons Hardware in 1940)
So S.H. Co would be a house brand name made by Hartford under a Crescent contract. Hartford Arms & Equipment made guns from 1925 to 1932 and was bought out by High Standard Co in 1932. Crescent was bought out by J. Stevens Arms around 1930, and they continued to make "house brand" guns for several years.