Omega Force was established in 2268 (Stardate 5462.1) as part of the Omega Directive developed by Starfleet Captain James T. Kirk following the destruction of subspace in the Lantaru sector by the Omega molecule.
As called for in the Omega Directive, Omega Force has carte blanche to destroy Omega by any means necessary, as under the Omega Directive, all priorities and regulations--even the Prime Directive--are temporarily rescinded.
As time went went on, and other threats to the Galaxy presented itself, the responsibilities of Omega Force increased. Now, they handle high-pressure situations where the presence of a fully-crewed starship is inappropriate.
The Omega Force is currently based out of Starbase 115 in the Balboa Sector. Due to the sensitive nature of the Omega Directive and the classified nature of Omega Force's very existence, its members are officially listed in Starfleet records as senior staff members aboard the USS Yagami, which is also based out of Starbase 115. (Star Trek: Omega Force)