Thank you for your replies.
On further investigation on the fitfreak forum and you tube there is a known fault on the keyless starter switch/button on the majority of Honda models from around 2013 to 2017. Apparently the fault can be fixed by cleaning the contacts in the starter switch/button.
Honda in America issued technical service bulletin 20-098 for the Fit giving the replacement of part number 35881-T5A-J02 with part number 35881-T5A-J03. They also extended the warrantee to 9 years. I do not know if this technical service bulletin applies in the UK.
On contacting a Honda spares department I was told part number 35881-T5A-J03 comprises of a starter switch and a module, are not sold separately and cost over £420 cannot remember if that included vat or not. There are numerous non-Honda starter switch/buttons available on E-bay in the £30 to £50 price range
On you tube there are numerous videos showing how to remove and clean the contacts on the starter switch/button on most of the Honda range including the Fit.
Taking the starter switch/button apart and cleaning the contacts is relatively easy and does not take long. The difficult, time consuming and nerve racking part is getting access to remove the starter button. You have to remove the bezel around the instrument cluster, a small part of the dashboard containing the air vents above the radio, the trim below the heating ventilation controls to allow the radio and other trim to be moved out of the way to remove the starter button. All of these are held in place with strong plastic clips which need lots of effort to remove and sound as if they going to snap. Reassembly is fairly straight forward.
I did this about 10 days ago and have not had the keyless warning light appear so cleaning the contacts has, so far fixed the fault.
I hope I have explained this clearly and not rambled on to much. If anyone wants and clarification please let me know.
Kind regards
FredS