Katherine
It is likely to be a Cap,Shoulder or Collar badge of the Lancashire Fusiliers.What you describe is the common shape of most British Fusilier Regimental badges that is of a "flaming grenade".The differences between the former Fusilier Regiments are in the detail of what may be inside the overall shape.In your case the presence of a Sphinx is a specific feature of the Lancashire Fusiliers.If it is a cap badge it is likely to be post 1898.
The Fusiliers Museum Lancashire is at Wellington Barracks,Bolton Road,Bury, LancashireBL8 2PL
Don