Fitting is much more expensive than the cost of the actual kitchen.
My kitchen cost under 5k, but I was 16k to get it fitted, but the kitchen needed taken back to bare skeleton due to the age/condition of the plasterboard and floor under the existing tiles.
Also, included in that cost was rewiring, connection of gas, new ceiling & lighting, laying new hardwood flooring throughout all of ground floor (5 rooms & hallway), new outdoor tap, new outdoor electrical sockets, removal of part of a wall, disposal of old kitchen and appliances, new skirting, removal & architrave and all painting/decorating.
Cut costs where you can with the type of units (drawer units cost heaps! As does any 'storage solutions' for corner cabinets), handles and worktops.
I sourced my unit handles from Etsy and my solid oak worktops from Worktop Express due to wanting specific sizes (longer than 3M) but it worked out cheaper than B&Q?
My range cooker was from AOL as they did a model I liked much cheaper than other places.
I used a lovely Polish fitter that I had used previously for my bathroom, who I knew did fantastic quality of workmanship. This can be better or worse than using a companies own 'fitters'?
Sometimes it works out cheaper, but if there are any issues or unexpected problems they're likely to say nothing & just put the units in anyway? But if you've got no wall/ceiling/floor issues then it may be the best option?
Basically, shop around, but expect to pay a lot in comparison to the price of the kitchen to have it fitted.
(I used to design kitchens, so I knew exactly what was required, which helped a lot.)