Kilgour was sold by its owner, private investment business KFS Group, for an undisclosed sum.
All the management including creative director Carlo Brandelli will remain at the tailoring business.
It is understood that JMH Group wants to expand the business internationally.
JMH Group supplies chemicals to the construction industry and is owned by James Hay, a former British Petroleum executive.
The group has also recently acquired gun company Ward Gunsmiths and horse racing business Upland Stables, as part of plans to create a luxury division.
Kilgour has two shops in Savile Row in London and wholesales to Harvey Nichols, Selfridges and Flannels.