The UNN Home fleet is the naval arm of the United Nations that operated in the inner Sol system. Seeing Mars as a major threat, the UNN kept a large number of ships in its navy as well as long-range nuclear missiles based on Earth and planetary defense railguns.
A Leonidas-class battleship from the Home Fleet was used to transport Chrisjen Avasarala to a VIP meeting with Jules-Pierre Mao. This ships commanding officer was ultimately loyal to Undersecretary Errinwright who ordered the ship to kill Avasarala. This was unsuccessful when the Rocinante crippled the engines of the battleship.
During the UN-MCR War the planetary defense railguns at Earth were used to fire upon and destroy 5 MCRN Stealth C Ballistic Missile Platforms. This was successful, but one of them fired a missile that the planetary defenses and Home Fleet failed to prevent from hitting the surface.
Later on three asteroids covered in stealth coating were able to get past the Home Fleet and planetary defenses undetected causing huge environmental damage and billions of deaths on the surface. These were coordinated by a Free Navy ship the Azure Dragon. With the risk of more asteroids to the already battered planet the Home Fleet was unable to move away from Earth to engage the enemy.
When the Azure Dragon was captured by the Rocinante the Home Fleet was able to move onto offensive operations launching the Ceres Offensive.